# property_assignments

structure_id: forget_abelian
property_id: finitary
type: functor
proof:
Since the forgetful functor $U_{\Grp} : \Grp \to \Set$ is finitary and conservative, it suffices to prove that the composition $U_{\Grp} \circ U_{\Ab,\Grp} : \Ab \to \Set$ is finitary. This is just the forgetful functor $U_{\Ab} : \Ab \to \Set$ and therefore finitary.


structure_id: forget_abelian
property_id: preserves finite coproducts
type: functor
proof:
The coproduct of two non-trivial cyclic groups in $\Ab$ is just their direct sum, which is finite, but the coproduct of these groups in $\Grp$, also called the free product, is an infinite group.


structure_id: forget_abelian
property_id: dominant
type: functor
proof:
Any subgroup of an abelian group is again abelian. Thus, if $G$ is any non-abelian group such as $S_3$, there cannot be an abelian group $A$ with a (split) monomorphism $G \hookrightarrow A$.


structure_id: forget_addition
property_id: conservative
type: functor
proof:
This is because even the composition $U_{\Mon} \circ U_{\Ring,\Mon} : \Ring \to \Set$, namely the forgetful functor $U_{\Ring} : \Ring \to \Set$, is conservative.


structure_id: forget_addition
property_id: finitary
type: functor
proof:
Since the forgetful functor $U_{\Mon} : \Mon \to \Set$ is finitary and conservative, it suffices to prove that the composition $U_{\Mon} \circ U_{\Ring,\Mon} : \Ring \to \Set$ is finitary. This is just the forgetful functor $U_{\Ring} : \Ring \to \Set$ and is therefore finitary.


structure_id: forget_addition
property_id: preserves reflexive coequalizers
type: functor
proof:
Since the forgetful functor $U_{\Mon} : \Mon \to \Set$ preserves reflexive coequalizers (by Theorem 2.5 at the <a href="https://ncatlab.org/nlab/show/reflexive+coequalizer" target="_blank">nLab</a>) and is conservative, it suffices to prove that the composition $U_{\Mon} \circ U_{\Ring,\Mon} : \Ring \to \Set$ preserves reflexive coequalizers. This is just the forgetful functor $U_{\Ring} : \Ring \to \Set$ and therefore preserves reflexive coequalizers (by loc.cit.).


structure_id: forget_addition
property_id: essentially injective
type: functor
proof:
See <a href="https://math.stackexchange.com/questions/4054295" target="_blank">MSE/4054295</a> for a collection of counterexamples.


structure_id: forget_addition
property_id: dominant
type: functor
proof:
A necessary condition for a monoid to be a retract of the multiplicative monoid of a ring is that it has an <a href="https://en.wikipedia.org/wiki/Absorbing_element" target="_blank">absorbing element</a>. Thus, for example, $(\IN,+,0)$ provides a counterexample. Remark: even a monoid with an absorbing element does not necessarily come from a ring; see <a href="https://math.stackexchange.com/questions/3075364" target="_blank">MSE/3075364</a>.


structure_id: forget_addition
property_id: preserves coequalizers
type: functor
proof:
Consider the two homomorphisms $f,g : \IZ[X] \rightrightarrows \IZ[X]$ defined by $f(X) = X^2$ and $g(X) = 0$. Their coequalizer is the ring $\IZ[X] / \langle X^2 \rangle$. In particular, it identifies $X^2 + X$ and $X$. We will prove that the coequalizer of $U(f), U(g) : U(\IZ[X]) \rightrightarrows U(\IZ[X])$ in $\Mon$ does not identify these elements.

The coequalizer in $\Mon$ is the quotient of $U(\IZ[X])$ by the smallest congruence relation satisfying $p(X^2) \sim p(0)$. Thus, two polynomials $f,g$ are equivalent iff there is a finite sequence $$f = f_0,f_1,\dotsc,f_n = g$$ where for each index $i$ we have $$\{f_i,f_{i+1}\} = \{p(X^2) q(X), p(0) q(X)\}$$ for some polynomials $p,q$ (that depend on the index).

Let $v : \IZ[X] \to \IN \cup \{\infty\}$ denote the $X$-adic valuation, returning the exponent of the largest power of $X$ dividing a polynomial. We claim that if $v(f)=1$ and $f \sim g$, then $v(g)=1$, and in that case $f/g \in \IQ(X^2)$. In fact, consider a step in the sequence $\{f_i,f_{i+1}\} = \{p(X^2) q(X), p(0) q(X)\}$ as above. If $p(0)=0$, then $p(0) q(X) = 0$ has valuation $\infty$, and $p(X^2) q(X)$ has valuation $\geq 2$. If $p(0) \neq 0$, then the valuations of $p(X^2) q(X)$ and $p(0) q(X)$ both equal the valuation of $q(X)$. Therefore, $v(f_i)=1 \iff v(f_{i+1})=1$. Moreover, in that case the ratio $f_i / f_{i+1}$ is either $p(X^2) / p(0)$ (which is well-defined) or $p(0) / p(X^2)$, both of which lie in $\IQ(X^2)$. From this, the claim follows.

Now we can conclude that $X^2+X$ and $X$ are not equivalent: both have valuation $1$, but their quotient is $X+1$, which is not contained in $\IQ(X^2)$.


structure_id: forget_group
property_id: finitary
type: functor
proof:
For every algebraic category the forgetful functor to the category of sets preserves filtered colimits.


structure_id: forget_group
property_id: preserves reflexive coequalizers
type: functor
proof:
This follows from Theorem 2.5 at the <a href="https://ncatlab.org/nlab/show/reflexive+coequalizer" target="_blank">nLab</a>.


structure_id: forget_group
property_id: dominant
type: functor
proof:
There is no group $G$ with a split monomorphism $\varnothing \hookrightarrow U(G)$ since $U(G)$ is non-empty.


structure_id: forget_group
property_id: preserves coequalizers
type: functor
proof:
The coequalizer of $1,2 : \IZ \rightrightarrows \IZ$ is the trivial group in $\Grp$, but the coequalizer of these maps in $\Set$ is infinite.


structure_id: forget_group_pointed
property_id: right adjoint
type: functor
proof:
A left adjoint can be constructed by mapping a pointed set $(X,x_0)$ to the free group $F(X)$ modulo $x_0 = 1$.


structure_id: forget_group_pointed
property_id: finitary
type: functor
proof:
Since the forgetful functor $U_{\Set_*} : \Set_* \to \Set$ is finitary and conservative, it suffices to prove that the composition $U_{\Set_*} \circ U_{\Grp,\Set_*} : \Grp \to \Set$ is finitary. This is just the forgetful functor $U_{\Grp} : \Grp \to \Set$ and therefore finitary.


structure_id: forget_group_pointed
property_id: preserves reflexive coequalizers
type: functor
proof:
It suffices to prove that $U_{\Grp} : \Grp \to \Set$ preserves them, which follows from Theorem 2.5 at the <a href="https://ncatlab.org/nlab/show/reflexive+coequalizer" target="_blank">nLab</a>.


structure_id: forget_group_pointed
property_id: essentially surjective
type: functor
proof:
Let $(X,x_0)$ be a pointed set. If $X$ is finite, we can endow $X$ with a cyclic group structure in which $x_0$ is the identity element. If $X$ is infinite, there is a bijection between $X$ and the set $P_{<\aleph_0}(X)$ of finite subsets of $X$, and we may assume that $x_0$ is mapped to $\varnothing$. Since $P_{<\aleph_0}(X)$ carries a group structure with identity element $\varnothing$ (it is the underlying set of the vector space $\IF_2^{\oplus X}$), $X$ also carries a group structure with identity element $x_0$.


structure_id: forget_group_pointed
property_id: preserves coequalizers
type: functor
proof:
The coequalizer of $1,2 : \IZ \rightrightarrows \IZ$ is the trivial group in $\Grp$, but the coequalizer of these maps in $\Set_*$ is infinite.


structure_id: forget_group_pointed
property_id: right-invertible
type: functor
proof:
Assume that there is a functor $G : \Set_* \to \Grp$ with $U \circ G \cong \id_{\Set_*}$, where $U : \Grp \to \Set_*$ is the forgetful functor. By using transport of structure, we can even assume that $U \circ G = \id_{\Set_*}$. Consider the set $X = \{x_0,x_1,x_2\}$ and the pointed set $(X,x_0)$. It carries a group structure with identity element $x_0$ such that every pointed map $(X,x_0) \to (X,x_0)$ is a group homomorphism (namely, its image under $G$). A group of order $3$ is isomorphic to $C_3$. If $t$ is a generator of $C_3$, then the map $U(C_3) \to U(C_3)$, $1 \mapsto 1$, $t \mapsto t$, $t^2 \mapsto t$ is pointed, but not a group homomorphism.


structure_id: forget_inverses
property_id: right adjoint
type: functor
proof:
Every forgetful functor between algebraic categories is a right adjoint. This functor is right adjoint to the enveloping group functor $\Mon \to \Grp$.


structure_id: forget_inverses
property_id: left adjoint
type: functor
proof:
This functor is left adjoint to the functor $(-)^{\times} : \Mon \to \Grp$ that sends a monoid $M$ to its group of units $M^{\times} \coloneqq \{(a,b) \in M^2 : ab = ba = 1 \}$. In fact, if $M$ is a monoid and $G$ is a group, there is a bijection $\Hom(U_{\Grp,\Mon}(G),M) \to \Hom(G,M^{\times})$ given by mapping $\varphi$ to $g \mapsto (\varphi(g),\varphi(g^{-1}))$.


structure_id: forget_inverses
property_id: dominant
type: functor
proof:
Every submonoid of a group is cancellative. Thus, if $M$ is a non-cancellative monoid such as $(\IN,\cdot,1)$, there cannot be a group $G$ with a (split) monomorphism $M \hookrightarrow G$.


structure_id: forget_ring
property_id: dominant
type: functor
proof:
There is no ring $R$ with a split monomorphism $\varnothing \hookrightarrow U(R)$ since $U(R)$ is non-empty.


structure_id: forget_ring
property_id: preserves epimorphisms
type: functor
proof:
This follows from the classifications of epimorphisms in <a href="/category/Ring">$\Ring$</a> and in <a href="/category/Set">$\Set$</a>. Namely, the unique homomorphism $\IZ \to \IQ$ is an epimorphism in $\Ring$, but its underlying map of sets is not surjective, and hence no epimorphism in $\Set$.


structure_id: forget_ring
property_id: preserves coequalizers
type: functor
proof:
Consider the two ring homomorphisms $\IZ[X] \rightrightarrows \IZ[X,Y]$ defined by $X \mapsto X$ resp. $X \mapsto 0$. Their coequalizer in $\Ring$ (and also $\CRing$) is $\IZ[Y]$. Every multiple of $X$ becomes zero. However, the $\Set$-coequalizer does not identify $XY$ with $0$. In fact, polynomials in $U(\IZ[X,Y]) \setminus U(\IZ[X])$ are only equivalent to themselves.


structure_id: forget_topology
property_id: reflector
type: functor
proof:
The indiscrete topology defines a fully faithful functor $I : \Set \to \Top$ which is right adjoint to $U_{\Top}$.


structure_id: forget_topology
property_id: coreflector
type: functor
proof:
The discrete topology defines a fully faithful functor $D : \Set \to \Top$ which is left adjoint to $U_{\Top}$.


structure_id: forget_topology
property_id: conservative
type: functor
proof:
If $X$ is a set, the map $D(X) \to I(X)$, $x \mapsto x$ is bijective and continuous (where $D(-)$ denotes the discrete topological space and $I(-)$ the indiscrete topological space), but not an isomorphism (unless $X$ has at most one element).


structure_id: forget_vector
property_id: dominant
type: functor
proof:
There is no vector space $V$ with a split monomorphism $\varnothing \hookrightarrow U(V)$ since $U(V)$ is non-empty.


structure_id: forget_vector
property_id: preserves coequalizers
type: functor
proof:
The coequalizer of $0,i_1 : K \to K^2$ in $\Vect$ is $p_2 : K^2 \to K$, but the coequalizer in $\Set$ is $(K \times K^*) \cup \{(0,0)\}$.


structure_id: free_group
property_id: preserves monomorphisms
type: functor
proof:
This can be deduced from the description of the elements of a free group, but here is an abstract argument: Split monomorphisms are preserved by any functor. The only injective maps in $\Set$ that are not split are $\varnothing \hookrightarrow X$ (for non-empty $X$), and $F(\varnothing) \to F(X)$ is injective since $F(\varnothing)$ is the trivial group.


structure_id: free_group
property_id: preserves coreflexive equalizers
type: functor
proof:
More generally, if $f,g : X \rightrightarrows Y$ are two injective maps with equalizer $E \hookrightarrow X$, then $F(E) \to F(X)$ is the equalizer of $F(f),F(g) : F(X) \rightrightarrows F(Y)$. In fact, we already know that $F(E) \to F(X)$ is injective. Now let $w \in F(X)$ be an element, and write it as a reduced word $w = x_1^{k_1} \cdots x_n^{k_n}$, meaning $x_i \in X$, $x_i \neq x_{i+1}$, $k_i \in \IZ \setminus \{0\}$. Since $f$ is injective, also $f(w) = f(x_1)^{k_1} \cdots f(x_n)^{k_n}$ is a reduced word. The same is true for $g(w)$. Hence, $f(w)=g(w)$ implies $f(x_i)=g(x_i)$ for every $i$, i.e. $x_i \in E$. Therefore, $w \in F(E)$.


structure_id: free_group
property_id: dominant
type: functor
proof:
If $G$ is a non-free group such as $\IZ/2$, there is no free group $F(X)$ with a (split) monomorphism $G \hookrightarrow F(X)$, since subgroups of free groups are free (Nielsen–Schreier theorem). Alternatively, use that $F(X)$ has no non-trivial elements of finite order.


structure_id: free_group
property_id: preserves equalizers
type: functor
proof:
Let $f,g : \{a,b\} \rightrightarrows \{c,d\}$ be the two constant maps, $f \equiv c$, $g \equiv d$. Their equalizer is empty, and the free group on that equalizer is trivial. Now consider the induced group homomorphisms $F(f), F(g) : F(\{a,b\}) \rightrightarrows F(\{c,d\})$ and observe that $a \cdot b^{-1}$ lies in the kernel of both homomorphisms, hence in their equalizer.


structure_id: free_group
property_id: cofinitary
type: functor
proof:
Consider the sets $X_n \coloneqq \{y,x_1,\dotsc,x_n\}$ for $n \geq 0$ with the canonical truncation maps $X_{n+1} \to X_n$, $x_i \mapsto x_i$, $x_{n+1} \mapsto y$, $y \mapsto y$. The limit $\lim_n X_n$ of this sequence is the set $L \coloneqq \{y,x_1,x_2,\dotsc\}$ with the canonical truncation maps $L \to X_n$. The canonical map $F(L) \to \lim_n F(X_n)$ is not surjective: For $n \geq 0$ define $w_n \in F(X_n)$ by $$w_n \coloneqq x_1 y^{-1} \cdots x_n y^{-1}.$$ Since $X_{n+1} \to X_n$ maps $w_{n+1} \mapsto w_n$, these elements yield a unique element $w \in \lim_n F(X_n)$. Notice that the word length of $w_n$ is unbounded in $n$. However, every element in $F(L)$ is contained already in $F(\{y,x_1,\dotsc,x_m\})$ for some $m$ and hence maps to an element with bounded word length. Hence, $w$ is not contained in the image.


structure_id: free_group
property_id: left-invertible
type: functor
proof:
Assume that there is a functor $H : \Grp \to \Set$ with $H \circ F \cong \id_\Set$. In particular, $H(1)= \varnothing$. Since for every $G \in \Grp$ there is a homomorphism $G \to 1$, there is a map $H(G) \to H(1) = \varnothing$, which forces $H(G) = \varnothing$. If we apply this to $G = F(X)$ for some non-empty set $X$, we obtain a contradiction.


structure_id: group_units
property_id: coreflector
type: functor
proof:
This functor is right adjoint to the forgetful functor $U_{\Grp,\Mon} : \Grp \hookrightarrow \Mon$, which is fully faithful.


structure_id: group_units
property_id: finitary
type: functor
proof:
There is a direct way to prove this, but here is an abstract argument: Since the forgetful functor $\Grp \to \Set$ is conservative and finitary, it suffices to prove that the composition $\Mon \to \Set$, which maps a monoid to its set of units, is finitary. This functor is represented by the monoid $\langle x,y : xy = yx = 1 \rangle \cong \IZ$. It has a finite presentation and hence is finitely presentable in the categorical sense (Corollary 3.13 in <a href="https://ncatlab.org/nlab/show/Locally+Presentable+and+Accessible+Categories" target="_blank">Adamek-Rosicky</a>).


structure_id: group_units
property_id: preserves coproducts
type: functor
proof:
We need to prove that for every family of monoids $(M_i)_{i \in I}$ the canonical map $$\textstyle \coprod_{i \in I} M_i^{\times} \to \bigl(\coprod_{i \in I} M_i\bigr)^{\times}$$ is an isomorphism. Notice that coproducts of monoids and groups have the same element structure; they are free products. From this, it follows immediately that the map is injective. For surjectivity, assume that $x = x_1 \cdots x_n$ and $y = y_1 \cdots y_m$ are reduced words in $\coprod_{i \in I} M_i$ with $xy = 1$ and $yx = 1$. We will prove by induction on $n$ that $n = m$ and that each $x_j$ is a unit in the corresponding monoid $M_{i_j}$.
If $n = 0$, i.e. $x = 1$, then we have $y = 1$, so $m = 0$, and the claim is true.
Let $n \geq 1$. The product $xy= x_1 \cdots x_n y_1 \cdots y_m$ would be in reduced form if $x_n$ and $y_1$ belong to different monoids; likewise, if they belong to the same monoid but $x_n y_1 \neq 1$. In that case, its free product length would be $n + m \geq 1$, so it could not evaluate to $1$. Thus, $x_n$ and $y_1$ necessarily belong to the same monoid, say $M_k$, and satisfy $x_n y_1 = 1$ in $M_k$. In particular, this implies that $m \geq 1$.
Next, consider the shorter elements $\tilde{x} \coloneqq x_1 \cdots x_{n-1}$ and $\tilde{y} \coloneqq y_2 \cdots y_m$, which are still in reduced form. Since $x_n y_1 = 1$, it follows that $1 = xy = \tilde{x} \tilde{y}$. On the other hand, $$\tilde{y}\tilde{x} = (x_n y_1) \tilde{y} \tilde{x} (x_n y_1)  = x_n (y x) y_1 = x_n y_1 = 1.$$ Thus, by the induction hypothesis applied to the pair $(\tilde{x},\tilde{y})$, all $x_1,\dotsc,x_{n-1}$ are units and $n-1 = m-1$, hence $n = m$. Moreover, $$1 = y x = y_1 (\tilde{y} \tilde{x}) x_n = y_1 x_n,$$ which shows that $1 = y_1 x_n$ in $M_k$. Combined with $x_n y_1 = 1$, this shows that $x_n$ is a unit as well.


structure_id: group_units
property_id: essentially injective
type: functor
proof:
If $M$ is the free monoid on one generator (i.e. $\IN$ w.r.t. addition), then $M^{\times}$ is trivial, just as the group of units of the trivial monoid, but $M$ is not trivial.


structure_id: group_units
property_id: faithful
type: functor
proof:
If $M$ is the free monoid on one generator (i.e. $\IN$ w.r.t. addition), then $M^{\times}$ is trivial, but $M$ is not. Hence, the identity $M \to M$ and the trivial homomorphism $M \to M$ have the same image under the functor, but they are not equal.


structure_id: group_units
property_id: full
type: functor
proof:
Let $M$ be a monoid and $G \subseteq M$ be its group of units. Then the identity $M^{\times} \to G^{\times} = G$ is in the image of the functor iff there is a homomorphism $M \to G$ that extends the identity on $G$, i.e. a retraction of $G \subseteq M$. If $M$ has an absorbing element, then its image in $G$ would be an absorbing element, forcing $G$ to be trivial. However, the monoid $(\IZ,\cdot,1)$, for example, has an absorbing element and a non-trivial group of units $\{\pm 1\}$.


structure_id: group_units
property_id: preserves regular epimorphisms
type: functor
proof:
Consider the monoids $M = \langle x : x^3 = x \rangle$ and $N = \langle y : y^2 = 1 \rangle$. Thus, $M$ has order $3$ and consists of $1,x,x^2$, and $N$ has order $2$ (and is even a group) with the elements $1,y$. We have a unique homomorphism $f : M \to N$ defined by $f(x)=y$, which is surjective. Since $M^{\times}$ is trivial and $N^{\times} \cong N$, the induced map $f^{\times} : M^{\times} \to N^{\times}$ is not surjective.


structure_id: indiscrete_topology
property_id: preserves coequalizers
type: functor
proof:
This follows easily from the description of coequalizers in $\Set$ and $\Top$, and the fact that if $p : X \to Y$ is a surjective map, then $I(p) : I(X) \to I(Y)$ is a quotient map.


structure_id: indiscrete_topology
property_id: finitary
type: functor
proof:
Let $X$ be a filtered diagram of sets. We need to prove that the filtered colimit of spaces $\colim_i I(X_i)$ is indiscrete. If $X_i$ is empty for all $i$, this is trivial. Otherwise, by considering a coslice of the index category, we may assume that every $X_i$ is non-empty. A topological space $Y$ is indiscrete iff there are at most two continuous maps $Y \to S$, where $S$ is the Sierpinski space; there are exactly two when $Y$ is non-empty. We have $$\textstyle \Hom(\colim_i I(X_i),S) \cong \lim_i \Hom(I(X_i),S) \cong \lim_i \{\varnothing, X_i\}.$$ For every $i \to j$ the map $I(X_i) \to I(X_j)$ pulls back $\varnothing \mapsto \varnothing$ and $X_j \mapsto X_i$. Thus, the limit has exactly two elements.


structure_id: indiscrete_topology
property_id: preserves finite coproducts
type: functor
proof:
If $X$ and $Y$ are two non-empty sets, the canonical continuous bijection $I(X) + I(Y) \to I(X + Y)$ is not an isomorphism, since $I(X) + I(Y)$ is not indiscrete; for instance, the subset $X$ is open.


structure_id: indiscrete_topology
property_id: dominant
type: functor
proof:
A subspace of an indiscrete space is again indiscrete. Hence, if $Y$ is any non-indiscrete space such as the discrete space $\IN$, there is no set $X$ with a regular (or even split) monomorphism $Y \hookrightarrow I(X)$.


structure_id: modulo-p
property_id: preserves products
type: functor
proof:
For a product of a family of abelian groups $(A_i)_{i \in I}$ it is easy to check that $$\textstyle p \prod_{i \in I} A_i = \prod_{i \in I} p A_i$$ as subgroups of $\prod_{i \in I} A_i$. Hence, $$\textstyle \prod_{i \in I} A_i / p \prod_{i \in I} A_i = \prod_{i \in I} A_i / \prod_{i \in I} p A_i \cong \prod_{i \in I} A_i / p A_i.$$


structure_id: modulo-p
property_id: dominant
type: functor
proof:
The essential image consists precisely of the elementary abelian $p$-groups, equivalently, the vector spaces over $\IF_p$, and these are closed under retracts. For example, $\IZ$ is not contained in it.


structure_id: modulo-p
property_id: faithful
type: functor
proof:
We have $\IQ / p \IQ \cong 0$, but $\IQ \not\cong 0$. Thus, faithfulness is failing for $0, \id_{\IQ} : \IQ \rightrightarrows \IQ$.


structure_id: modulo-p
property_id: full
type: functor
proof:
Let $\IZ_{(p)}$ denote the additive group of $p$-local numbers, i.e. the localization of $\IZ$ at the prime ideal $\langle p \rangle$. The map from $\Hom(\IZ_{(p)},\IZ)$ to $\Hom(\IZ_{(p)} / p \IZ_{(p)} ,\IZ / p)$ is not surjective: The abelian group $\HomInternal(\IZ_{(p)},\IZ)$ is trivial: For a prime $q \neq p$ every element of $\IZ_{(p)}$ is $q^\infty$-divisible, but $0$ is the only $q^\infty$-divisible integer. However, $\IZ_{(p)} / p \IZ_{(p)} \cong \IZ/p$ shows that $\HomInternal(\IZ_{(p)} / p \IZ_{(p)} ,\IZ / p)$ is isomorphic to $\IZ/p$.


structure_id: modulo-p
property_id: cofinitary
type: functor
proof:
Let $q$ be be an integer which is $\geq 2$ and coprime to $p$. Consider the sequence $$\cdots \xrightarrow{q} \IZ \xrightarrow{q} \IZ  \xrightarrow{q} \IZ.$$ Its limit is $0$ since each entry needs to be $q^\infty$-divisible. However, the sequence modulo $p$ $$\cdots \xrightarrow{q} \IZ/p \xrightarrow{q} \IZ/p  \xrightarrow{q} \IZ/p$$ consists entirely of isomorphisms, so that its limit is $\IZ/p$.


structure_id: monoid_ring
property_id: left adjoint
type: functor
proof:
This functor is left adjoint to the forgetful functor $U_{\Ring,\Mon} : \Ring \to \Mon$, which maps a ring to its underlying multiplicative monoid.


structure_id: monoid_ring
property_id: conservative
type: functor
proof:
Assume that $f : M \to N$ is a homomorphism of monoids such that the induced ring homomorphism $\IZ[f] : \IZ[M] \to \IZ[N]$ is bijective. Since the natural map $N \to \IZ[N]$ is injective, it follows immediately that $f$ is injective. Now let $n \in N$ and choose a preimage $\sum_m z_m \cdot m$ in $\IZ[M]$. Thus, $n = \sum_m z_m \cdot f(m)$. If $f(m) \neq n$ for all $m$, this would contradict the linear independence of $N$ in $\IZ[N]$. Thus, $n$ lies in the image of $f$.


structure_id: monoid_ring
property_id: preserves monomorphisms
type: functor
proof:
Let $f : M \to N$ be an injective monoid homomorphism. To show that the ring homomorphism $\IZ[f] : \IZ[M] \to \IZ[N]$ is injective, take an element in the kernel, say $\sum_{m \in M} z_m \cdot m$. Then $\sum_{m \in M} z_m \cdot f(m) = 0$ in $\IZ[N]$. Since the $f(m) \in N$ are pairwise distinct and linearly independent over $\IZ$, it follows that $z_m = 0$ for every $m \in M$, proving the claim.


structure_id: monoid_ring
property_id: preserves coreflexive equalizers
type: functor
proof:
Let $f,g : M \rightrightarrows N$ be homomorphisms of monoids with a common retraction $r : N \to M$, i.e. $r f = r g = \id_M$. Let $E = \{m \in M : f(m) = g(m)\}$ be the equalizer of $f$ and $g$. Consider an element $x \in \IZ[M]$ that is equalized by $\IZ[f], \IZ[g]$; we need to show that $x \in \IZ[E]$ (which is a subring of $\IZ[M]$, as we have seen before). Write $x = \sum_m z_m \cdot m$. Then $\sum_m z_m \cdot f(m) = \sum_m z_m \cdot g(m)$. If $n \in N$, comparing coefficients on both sides shows that $\sum_{m \in f^*(n)} z_m = \sum_{m \in g^*(n)} z_m$ in $\IZ$. Since $f$ and $g$ are injective, the fibers have at most one element. Now let $m \in M$ with $z_m \neq 0$. Let $n \coloneqq f(m)$. By the coefficient equation above, we deduce that $g^*(n)$ must be non-empty. Choose $m' \in M$ with $g(m') = n$. By applying $r$ to $f(m) = g(m')$, we get $m = m'$. Thus, $m \in E$. This shows that $x \in \IZ[E]$.


structure_id: monoid_ring
property_id: full
type: functor
proof:
Take $M = \langle X \rangle$ and $N = \{1\}$, the free monoids of ranks $1$ and $0$, respectively. Their monoid rings are $\IZ[X]$ (the polynomial ring) and $\IZ$. The only ring homomorphism $\IZ[X] \to \IZ$ in the image of the functor is $X \mapsto 1$. But there are many more, such as $X \mapsto 0$.


structure_id: monoid_ring
property_id: dominant
type: functor
proof:
Every monoid ring is torsion-free as an abelian group (in fact, the underlying additive group is free abelian), and a subring of a torsion-free ring is again torsion-free. In particular, the ring $\IZ/2$ is not a retract of a monoid ring.


structure_id: monoid_ring
property_id: preserves equalizers
type: functor
proof:
Consider the free monoids $M = \langle a,b \rangle$, $N = \langle c \rangle$ and the homomorphisms $f,g : M \rightrightarrows N$ defined by $f(a)=c$, $f(b)=1$ and $g(a)=1$, $g(b)=c$. Then $a$ and $b$ are not contained in the equalizer $E$ of $f$ and $g$. More generally, $E$ contains exactly the words in $a,b$ in which $a$ and $b$ occur the same number of times. The element $a+b \in \IZ[M]$ is equalized by $\IZ[f], \IZ[g]$ since $\IZ[f]$ maps it to $c+1$, while $\IZ[g]$ maps it to $1+c$. But it is not contained in $\IZ[E]$.


structure_id: monoid_ring
property_id: cofinitary
type: functor
proof:
Consider the homomorphisms of commutative additive monoids $\IN^{n+1} \to \IN^n$ defined by $(k_1,\dotsc,k_{n+1}) \mapsto (k_1,\dotsc,k_n)$ for $n \geq 1$. Their limit is the power $\IN^\omega$. The natural ring homomorphism $\alpha : \IZ[\IN^\omega] \to \lim_n \IZ[\IN^n]$ is not surjective: The ring $\IZ[\IN^n]$ identifies with the polynomial ring $\IZ[X_1,\dotsc,X_n]$, and the transition maps are given by $X_{n+1} \mapsto 1$. The ring $\IZ[\IN^\omega]$ can be seen as a variant of the polynomial ring $\IZ[X_1,X_2,\dotsc]$ in which monomials are allowed to contain infinitely many variables, but a polynomial is still just a finite linear combination of such monomials. The homomorphism $\alpha$ inserts $1$ in the unused variables: $\alpha(X_1^{k_1} X_2^{k_2} \cdots)_n = X_1^{k_1} X_2^{k_2} \cdots X_n^{k_n}$. For every element in the image of $\alpha$ there is some $N \in \IN$ such that every entry in $\IZ[X_1,\dotsc,X_n]$ is a sum of at most $N$ monomials. Now consider the element $f \in \lim_n \IZ[X_1,\dotsc,X_n]$ defined by $f_n = \sum_{i=1}^n (X_i - 1)$. It is indeed contained in the limit, since $X_{n+1} \mapsto 1$ turns $f_{n+1}$ into $f_n$. Since $f_n$ is a sum of $n+1$ monomials, $f$ is not contained in the image of $\alpha$.


structure_id: p-torsion
property_id: finitary
type: functor
proof:
This follows abstractly from the fact that $\IZ/p$ is finitely presentable. We nevertheless give a direct proof. Let $(A_i)$ be a filtered diagram of abelian groups. We need to show that the canonical map $$\colim_i T_p(A_i) \to T_p(\colim_i A_i)$$ is an isomorphism.
Injectivity: Every element of $\colim_i T_p(A_i)$ is induced by an element in some $T_p(A_i)$. Assume it lies in the kernel, i.e. it becomes zero in $\colim_i A_i$. Then there exists some $i \to j$ such that its image in $A_j$ is zero. Hence its image in $T_p(A_j)$ is zero, and we are done.
Surjectivity: Assume we have a $p$-torsion element in $\colim_i A_i$. It has a preimage in some $A_i$. When we multiply it with $p$, it becomes zero in the colimit. This means that there exists some $i \to j$ such that this multiple becomes zero in $A_j$. Therefore, the image in $A_j$ is a $p$-torsion element. It provides the desired preimage.


structure_id: p-torsion
property_id: faithful
type: functor
proof:
We have $T_p(\IZ) = 0$, but $\IZ \not\cong 0$. Thus, faithfulness is failing for $0, \id_{\IZ} : \IZ \rightrightarrows \IZ$.


structure_id: p-torsion
property_id: full
type: functor
proof:
The canonical map from $\Hom(\IZ/p^2,\IZ/p)$ to $\Hom(T_p(\IZ/p^2),T_p(\IZ/p)) = \Hom(p \IZ/p^2, \IZ/p)$ is not surjective: The homomorphism $[p] \mapsto [1]$ is not contained in its image.


structure_id: p-torsion
property_id: preserves epimorphisms
type: functor
proof:
The surjective homomorphism $\IZ \to \IZ/p$ is mapped to the homomorphism $0 \to \IZ/p$, which is not surjective.


structure_id: pi_0
property_id: preserves products
type: functor
proof:
See the <a href="https://ncatlab.org/nlab/show/connected+space#pathcomponents_functor" target="_blank">nLab</a>.


structure_id: pi_0
property_id: preserves epimorphisms
type: functor
proof:
If $X \to Y$ is an epimorphism in $\Top$, this means that $U(X) \to U(Y)$ is surjective. Hence, also $U(X)/{\sim} = \pi_0(X) \to \pi_0(Y) = U(Y)/{\sim}$ is surjective.


structure_id: pi_0
property_id: conservative
type: functor
proof:
The inclusion map $\{0\} \hookrightarrow [0,1]$ induces an isomorphism $\pi_0(\{0\}) \to \pi_0([0,1])$, but is not an isomorphism itself.


structure_id: pi_0
property_id: full
type: functor
proof:
Take two totally disconnected spaces $X,Y$ which admit a map $f : U(X) \to U(Y)$ that is not continuous, for example $X = Y = \IQ$ with the usual topology, $f(0)=1$ and $f(x)=0$ for $x \neq 0$. Then $f$ induces a map $\pi_0(X) \cong U(X) \to U(Y) \cong \pi_0(Y)$ that is not induced by a continuous map $X \to Y$.


structure_id: pi_0
property_id: preserves regular monomorphisms
type: functor
proof:
The functor $\pi_0$ maps the embedding $\{0,1\} \hookrightarrow [0,1]$ to the (unique) map $\{\{0\},\{1\}\} \to \{[0,1]\}$, which is not injective.


structure_id: pi_0
property_id: cofinitary
type: functor
proof:
Consider the spaces $X_n = [n,\infty) \subseteq \IR$ and the inclusion maps $X_{n+1} \hookrightarrow X_n$. Then $\lim_n X_n = \varnothing$ and hence $\pi_0(\lim_n X_n) = \varnothing$. However, each $\pi_0(X_n) = \{X_n\}$ is a singleton, so that also $\lim_n \pi_0(X_n)$ is a singleton.


structure_id: pi_0
property_id: finitary
type: functor
proof:
Let $A_n = \{1/k : k \geq n\}$ for $n \geq 1$. Let $X_n$ be the interval $[0,1]$ equipped with the topology generated by the Euclidean topology and all the singletons $\{a\}$ for $a \in A_n$. The identity maps $X_n \to X_{n+1}$ are continuous since $A_{n+1} \subseteq A_n$, and we have continuous identity maps $X_n \to [0,1]$. Using $\bigcap_{n \geq 1} A_n = \varnothing$, one can show that $\operatorname{colim}_n X_n = [0,1]$ with the Euclidean topology, which is path connected. In particular, $[0]=[1]$ in $\pi_0(\operatorname{colim}_n X_n)$. However, this equation does not hold in $\operatorname{colim}_n \pi_0(X_n)$. Otherwise, there is some $n$ such that it holds in $\pi_0(X_n)$, i.e. there is a path $\gamma : [0,1] \to X_n$ with $\gamma(0)=0$ and $\gamma(1)=1$. By the intermediate value theorem applied to the composition with the map $X_n \to [0,1]$, the element $1/n$ lies in its image. But $\{1/n\}$ is open and closed in $X_n$. Hence, its preimage is $[0,1]$, which means that the path is constant. This is a contradiction.


structure_id: pi_1
property_id: preserves products
type: functor
proof:
This is straightforward to check. A reference for binary products is Theorem III.2.6 in Bredon's <a href="https://doi.org/10.1007/978-1-4757-6848-0" target="_blank">Topology and Geometry</a>. The general case works exactly the same.


structure_id: pi_1
property_id: right-invertible
type: functor
proof:
The classifying space $BG$ of a group $G$ is functorial in $G$ and satisfies $\pi_1(BG) \cong G$; see Hatcher's <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html" target="_blank">Algebraic Topology</a>, Example 1B.7.


structure_id: pi_1
property_id: conservative
type: functor
proof:
The inclusion map $(\{0\},0) \hookrightarrow ([0,1],0)$ induces an isomorphism of groups $\pi_1(\{0\},0) \to \pi_1([0,1],0)$ (both groups are trivial), but is not an isomorphism itself.


structure_id: pi_1
property_id: faithful
type: functor
proof:
Any two pointed continuous maps $(\IR,0) \to (\IR,0)$, such as $x \mapsto \pm x$, have the same image under $\pi_1$ since $\pi_1(\IR,0)$ is the trivial group. More generally, homotopic pointed maps have the same image under $\pi_1$.


structure_id: pi_1
property_id: essentially injective
type: functor
proof:
We have $\pi_1(\{0\},0) \cong \pi_1([0,1],0)$ (both groups are trivial), but $(\{0\},0) \not\cong ([0,1],0)$.


structure_id: pi_1
property_id: preserves regular monomorphisms
type: functor
proof:
The functor $\pi_1$ maps the embedding $(S^1,1) \hookrightarrow (\IC,1)$ to the trivial group homomorphism $\IZ \to \{1\}$, which is not injective.


structure_id: pi_1
property_id: preserves regular epimorphisms
type: functor
proof:
The functor $\pi_1$ maps the regular epimorphism $(\IR,0) \to (S^1,1)$, $t \mapsto e^{2 \pi i t}$, to the trivial group homomorphism $\{1\} \to \IZ$, which is not surjective.


structure_id: pi_1
property_id: preserves binary coproducts
type: functor
proof:
See <a href="https://math.stackexchange.com/questions/3991443" target="_blank">MSE/3991443</a> or <a href="https://math.stackexchange.com/questions/2788471" target="_blank">MSE/2788471</a> for counterexamples. See <a href="https://math.stackexchange.com/questions/4625240" target="_blank">MSE/4625240</a> for conditions which binary products are preserved.


structure_id: pi_1
property_id: finitary
type: functor
proof:
Consider the sequence of pointed spaces $$(S^1,1) \xrightarrow{f} (S^1,1) \xrightarrow{f} (S^1,1) \xrightarrow{f} \cdots \tag{S}$$ where $f(z) \coloneqq z^2$. The induced sequence of fundamental groups is isomorphic to $$\IZ \xrightarrow{2} \IZ \xrightarrow{2} \IZ \xrightarrow{2} \cdots,$$ whose colimit is $\IZ[1/2]$, the group of dyadic rationals. In contrast, since $f$ is a surjective open map, the colimit $(X,x_0)$ of (S) is a quotient of $(S^1,1)$, namely $(S^1 / G,[1])$, where $G \subseteq S^1$ is the subgroup of elements $a \in S^1$ satisfying $a^{2^n}=1$ for some $n \in \IN$. The quotient $S^1 / G$ is indiscrete: any non-empty open subset pulls back to a non-empty open subset of $S^1$ that is $G$-invariant, and this must be all of $S^1$ since $G \subseteq S^1$ is dense. Now it is easy to check that the fundamental group of an indiscrete pointed space is trivial. This proves that $\pi_1$ does not preserve the colimit of (S).


structure_id: power_set_contravariant
property_id: representable
type: functor
proof:
This is because there are natural bijections $P(X) \cong \Hom(X,2)$, sending a subset to its characteristic function.


structure_id: power_set_contravariant
property_id: conservative
type: functor
proof:
Assume that $f : X \to Y$ is a map such that $f^* : P(Y) \to P(X)$ is bijective. For $y \in Y$ it follows that $f^*(\{y\}) \neq f^*(\varnothing) = \varnothing$, which means that $y$ has an $f$-preimage. That is, $f$ is surjective. Next, if $x,x' \in X$ satisfy $f(x)=f(x')$, choose $T \in P(Y)$ with $f^*(T) = \{x\}$. Then $f(x') = f(x) \in T$, hence $x' \in f^*(T) = \{x\}$. This shows that $f$ is injective.


structure_id: power_set_contravariant
property_id: right adjoint
type: functor
proof:
The functor $P_{\forall} : \Set^{\op} \to \Set$ is right adjoint to $P_{\forall}^{\op} : \Set \to \Set^{\op}$ since $$\Hom_{\Set}(X,P(Y)) \cong P(X \times Y) \cong \Hom_{\Set}(Y,P(X)) = \Hom_{\Set^{\op}}(P(X),Y).$$


structure_id: power_set_contravariant
property_id: preserves reflexive coequalizers
type: functor
proof:
This is contained in the proof of Theorem 2.2.7 (monadicity of $P_{\forall}$ for an elementary topos) in <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a>, but we include a direct proof here for the sake of completeness, and also since we want the system to deduce monadicity automatically.
Let $f,g : X \rightrightarrows Y$ be two maps with a common retraction $r : Y \to X$, i.e. $rf = rg = \id_X$. Let $i : E \hookrightarrow X$ be their equalizer. We need to show that $i^* : P(X) \to P(E)$ is a coequalizer of $f^*,g^* : P(Y) \rightrightarrows P(X)$. First, $i^*$ is surjective since $i$ is injective, which implies that $A = i^*(i_*(A))$ for all $A \subseteq E$. Hence, $i^*$ is the quotient of $P(X)$ by the equivalence relation $\sim_1$ defined by $A \sim_1 B \iff i^*(A) = i^*(B)$, i.e. $A \cap E = B \cap E$. The coequalizer of $f^*,g^*$ is the quotient of $P(X)$ by the smallest equivalence relation $\sim_2$ satisfying $f^*(C) \sim_2 g^*(C)$ for all $C \subseteq Y$. We need to show that these equivalence relations coincide.
(1) $\sim_2$ is contained in $\sim_1$: If $C \subseteq Y$, then $f \circ i = g \circ i$ implies $i^*(f^*(C)) = i^*(g^*(C))$, so that $f^*(C) \sim_1 g^*(C)$.
(2) $\sim_1$ is contained in $\sim_2$: Let $A,B \subseteq X$ be subsets with $i^*(A) = i^*(B)$. We claim that $g^*(f_*(A)) = g^*(f_*(B))$. In fact, if $x \in g^*(f_*(A))$, then $g(x) = f(a)$ for some $a \in A$. Applying $r : Y \to X$ shows that $x = a$. Thus, $a \in A \cap E = B \cap E$, and we get $x \in g^*(f_*(B))$. The claim now implies $$A = f^*(f_*(A)) \sim_2 g^*(f_*(A)) = g^*(f_*(B)) \sim_2 f^*(f_*(B)) = B.$$


structure_id: power_set_contravariant
property_id: full on isomorphisms
type: functor
proof:
If $f : X \to Y$ is any map, then $f^* : P(Y) \to P(X)$ preserves the empty set (in fact, all unions). Thus, it suffices to take any bijective map $P(X) \to P(X)$ that does not preserve $\varnothing$, such as the transposition of $\varnothing$ and $X$ (where $X$ is non-empty).


structure_id: power_set_contravariant
property_id: preserves initial objects
type: functor
proof:
In fact, the initial object does not even lie in the essential image since $P(X) \neq \varnothing$ for all $X$.


structure_id: power_set_contravariant
property_id: dominant
type: functor
proof:
There is no split monomorphism $\varnothing \hookrightarrow P(X)$ for any set $X$ since $P(X)$ is non-empty.


structure_id: power_set_contravariant
property_id: preserves coequalizers
type: functor
proof:
Let $X = \{0,1\}$ and $Y = \{a,b,c\}$. Define maps $f,g : Y \rightrightarrows X$ by $$\begin{align*} f(a)=0,\, f(b)=0,\, f(c)=1; \\ g(a)=0,\, g(b)=1,\, g(c)=0. \end{align*}$$ Their equalizer is $E = \{a\}$, so that $P(E)$ has $2$ elements. We compute $$\begin{align*} f^*(\{0\})=\{a,b\},\quad & g^*(\{0\})=\{a,c\},\\ f^*(\{1\})=\{c\},\quad & g^*(\{1\})=\{b\}. \end{align*}$$ Thus the coequalizer of $f^*,g^*$ is obtained from $P(Y)$ by imposing the two relations $\{a,b\} \sim \{a,c\}$ and $\{c\} \sim \{b\}$. But then it as $8-2 = 6$ elements.


structure_id: power_set_contravariant
property_id: finitary
type: functor
proof:
Consider the sequence of truncation maps $\cdots \twoheadrightarrow \{0,1\}^2 \twoheadrightarrow \{0,1\}^1$. Its limit in $\Set$ (i.e. colimit in $\Set^{\op}$) is $\{0,1\}^{\IN}$, which is uncountable, so that $P(\{0,1\}^{\IN})$ is also uncountable. But the colimit of the induced diagram $P(\{0,1\}^1) \hookrightarrow P(\{0,1\}^2) \hookrightarrow \cdots$ is countable since each set $P(\{0,1\}^n)$ is finite.


structure_id: power_set_contravariant
property_id: left-invertible
type: functor
proof:
Assume for a contradiction that there is a functor $F : \Set \to \Set^{\op}$ with (natural) isomorphisms $X \cong F(P(X))$. For any non-empty set $X$ there is a map $1 \to X$, which yields a map $F(X) \to F(1)$. Since $F(1) \cong F(P(0)) \cong 0$ is empty, it follows that also $F(X)$ is empty. Since this holds for every non-empty set, it also follows that $X \cong F(P(X)) \cong 0$, which is a contradiction.


structure_id: power_set_contravariant
property_id: essentially injective
type: functor
proof:
The property $P(X) \cong P(Y) \implies X \cong Y$ is undecidable in ZFC. It holds when the generalized continuum hypothesis is true, but it can fail otherwise. A more precise result is given by <a href="https://en.wikipedia.org/wiki/Easton's_theorem" target="_blank">Easton's Theorem</a>. See also <a href="https://mathoverflow.net/questions/17152" target="_blank">MO/17152</a>.


structure_id: power_set_covariant
property_id: faithful
type: functor
proof:
Let $f,g : X \rightrightarrows Y$ be two maps with $f_* = g_* : P(X) \rightrightarrows P(Y)$. Then $\{f(x)\} = f_*(\{x\}) = g_*(\{x\}) = \{g(x)\}$ and hence $f(x) = g(x)$ for all $x \in X$.


structure_id: power_set_covariant
property_id: conservative
type: functor
proof:
Assume that $f : X \to Y$ is a map such that $f_* : P(X) \to P(Y)$ is an isomorphism. There is some $A \subseteq X$ with $Y = f_*(A)$, this proves that $f$ is surjective. It is also injective: If $x,y \in X$ satisfy $f(x) = f(y)$, then $f_*(\{x\}) = f_*(\{y\})$, and hence $\{x\} = \{y\}$, i.e. $x = y$.


structure_id: power_set_covariant
property_id: preserves coreflexive equalizers
type: functor
proof:
Let $f,g : X \rightrightarrows Y$ be a coreflexive pair in $\Set$. Choose a common retraction $r : Y \to X$, so that $rf = rg = \id_X$. Let $E = \{x \in X : f(x)=g(x)\}$ be the usual equalizer in $\Set$. We must show that for every subset $A \subseteq X$, the equality $f_*(A) = g_*(A)$ implies $A \subseteq E$. To see this, let $a \in A$. Then $f(a)=g(a')$ for some $a' \in A$. Applying $r$ to both sides yields $a=a'$. Hence $f(a)=g(a)$, so $a \in E$.


structure_id: power_set_covariant
property_id: full on isomorphisms
type: functor
proof:
If $f : X \to Y$ is any map, then $f_* : P(Y) \to P(X)$ preserves the empty set (in fact, all unions). Thus, it suffices to take any bijective map $P(X) \to P(X)$ that does not preserve $\varnothing$, such as the transposition of $\varnothing$ and $X$ (where $X$ is non-empty).


structure_id: power_set_covariant
property_id: preserves reflexive coequalizers
type: functor
proof:
Consider the two maps $f,g : \IN \times \{0,1\} \rightrightarrows \IN$ defined by $f(n,i)=n$, $g(n,0)=n$, and $g(n,1)=n+1$. They form a reflexive pair with common section $n \mapsto (n,0)$. Their coequalizer is the quotient of $\IN$ by the equivalence relation generated by $n \sim n+1$, which collapses everything to a single point. Thus, the unique map $p : \IN \to \{\ast\}$ is a coequalizer of $f,g$. The induced map $p_* : P(\IN) \to P(\{\ast\}) = \{\varnothing,\{\ast\}\}$ sends $\varnothing$ to $\varnothing$ and every non-empty subset to $\{\ast\}$. In particular, all non-empty subsets of $\IN$ have the same image.
Next, we compute the coequalizer of $f_*, g_* : P(\IN \times \{0,1\}) \rightrightarrows P(\IN)$. Identify $P(\IN \times \{0,1\})$ with $P(\IN) \times P(\IN)$. Under this identification, $f_*$ sends a pair of subsets $(A,B)$ to $A \cup B$, while $g_*$ sends it to $A \cup (B+1)$.
The coequalizer is the quotient of $P(\IN)$ by the equivalence relation generated by $$A \cup B \sim A \cup (B+1).$$ This generated equivalence relation is the transitive closure of the symmetric closure. From this it follows that a finite subset is equivalent only to finite subsets. In particular, $\{0\}$ and $\IN$ are not equivalent, i.e. they do not have the same image in the coequalizer. Therefore, $p_*$ cannot be the coequalizer.


structure_id: power_set_covariant
property_id: preserves equalizers
type: functor
proof:
Any pair of distinct surjective maps $f,g : X \rightrightarrows Y$ provides a counterexample: Their equalizer $E$ is a proper subset of $X$, so that $P(E)$ cannot contain $X$. But $f_*(X) = Y = g_*(X)$ shows that $X$ is contained in the equalizer of $f_*,g_* : P(X) \rightrightarrows P(Y)$.


structure_id: power_set_covariant
property_id: finitary
type: functor
proof:
The filtered colimit $\IN = \bigcup_{n \geq 0} \IN_{\leq n}$ is not preserved by $P$, since $\bigcup_{n \geq 0} P(\IN_{\leq n})$ just consists of the <i>finite</i> subsets of $\IN$.


structure_id: power_set_covariant
property_id: cofinitary
type: functor
proof:
The canonical map $P(\lim_i X_i) \to \lim_i P(X_i)$ is typically not injective already for cardinality reasons: Consider for example the sequential diagram $X_n \coloneqq \{0,1\}^n$ with the truncation maps $X_{n+1} \to X_n$. The limit is $\{0,1\}^{\IN}$ with cardinality $c \coloneqq 2^{\aleph_0}$, so that its power set has cardinality $2^c$. However, the limit of the power sets $P(X_n)$ is a subset of a countable product of finite sets, hence has cardinality $\leq c$, which is strictly smaller than $2^c$.


structure_id: power_set_covariant
property_id: left-invertible
type: functor
proof:
Assume that there is a functor $F : \Set \to \Set$ with natural isomorphisms $$\alpha_X : X \to F(P(X)).$$ By the Yoneda Lemma, there is some element $u \in F(P(1))$ (where $1$ is the singleton set as usual) such that $\alpha_X$ is given by $$\alpha_X(x) = F(P(x : 1 \to X))(u).$$ The map $P(x) : P(1) \to P(X)$ is given by $\varnothing \mapsto \varnothing$ and $1 \mapsto \{x\}$. It factors through the map $e : X + 1 \to P(X)$ which maps $a \in X$ to $\{a\}$ and the unique element of $1$ to $\varnothing$. Since $\alpha_X$ is surjective, it follows that also $F(e) : F(X + 1) \to F(P(X))$ is surjective. Since $e$ is a split monomorphism, also $F(e)$ is a split monomorphism. Hence, $F(e)$ must be an isomorphism. But then $F(X+1) \cong F(P(X)) \cong X$ for all sets $X$. If $X$ is a finite non-empty set, we conclude $\card(F(X)) = \card(X) - 1$. But then $$\card(X) = \card(F(P(X))) = \card(P(X)) - 1 = 2^{\card(X)} - 1$$ yields a contradiction when $\card(X) > 1$.


structure_id: sequences_sets
property_id: monadic
type: functor
proof:
This is a special case of Theorem 1 in <a href="https://arxiv.org/abs/math/0407251" target="_blank">State monads and their algebras</a> by F. Metayer.


structure_id: sequences_sets
property_id: essentially injective
type: functor
proof:
There is an isomorphism $2^{\IN} \cong 2^{2 \cdot \IN} \cong 4^{\IN}$. There are also counterexamples for infinite sets, such as $\IN^{\IN} \cong 2^{\IN} \cong \IR \cong \IR^{\IN}$.


structure_id: sequences_sets
property_id: finitary
type: functor
proof:
If $X$ is any set, it is the colimit of its finite subsets (ordered by inclusion), and the canonical map $$\colim_{E \subseteq X \text{ finite}} E^{\IN} \hookrightarrow X^{\IN}$$ is the inclusion of the set of sequences with only finitely many values. Hence, when $X$ is infinite, it is not bijective.


structure_id: sequences_sets
property_id: preserves finite coproducts
type: functor
proof:
The natural map $X^{\IN} + Y^{\IN} \hookrightarrow (X+Y)^{\IN}$ is injective, but it is not surjective if $X$ and $Y$ are both non-empty: Take $x \in X$, $y \in Y$, then the sequence $(x,y,y,y,\dotsc)$ is not contained in the image.


structure_id: sequences_sets
property_id: preserves reflexive coequalizers
type: functor
proof:
Let $E \coloneqq \{(a,b) \in \IZ^2 : |a-b| \leq 1 \}$. We have two projection maps $p_1,p_2 : E \rightrightarrows \IZ$ with common section $a \mapsto (a,a)$. The coequalizer of $p_1,p_2$ is the quotient of $\IZ$ by the equivalence relation generated by $a \sim a + 1$ for $a \in \IZ$. Thus, all integers become identified, and the unique map $\IZ \to \{\ast\}$ is a coequalizer of $p_1,p_2$. We will show that the coequalizer of ${p_1}^{\IN},{p_2}^{\IN}$ is much larger than $\{\ast\}^{\IN} \cong \{\ast\}$.
The set $E^{\IN}$ can be identified with the set of pairs of integer sequences $(a,b) \in \IZ^{\IN} \times \IZ^{\IN}$ such that $|a_n - b_n| \leq 1$ for all $n \in \IN$. We have ${p_1}^{\IN}(a,b)=a$ and ${p_2}^{\IN}(a,b)=b$. Hence, their coequalizer is the quotient of $\IZ^{\IN}$ by the equivalence relation generated by the condition $|a_n - b_n| \leq 1$ for all $n \in \IN$. This relation can be described explicitly: two sequences $a,b$ are equivalent if and only if their difference $a-b$ is bounded, i.e. $\sup_{n \in \IN} |a_n - b_n| < \infty$. Therefore, there are many non-equivalent sequences, for example $(0,0,0,\dotsc)$ and $(1,2,3,\dotsc)$.


structure_id: squaring_sets
property_id: conservative
type: functor
proof:
This is a straight forward calculation. Assume that $f : X \to Y$ is a map such that $f^2 : X^2 \to Y^2$ is bijective. For every $y \in Y$ there is some $(x_1,x_2) \in X^2$ with $(y,y) = (f(x_1),f(x_2))$, so that $y = f(x_1)$. This shows that $f$ is surjective. To show that $f$ is injective, suppose that $x_1,x_2 \in X$ satisfy $f(x_1)=f(x_2)$. Then $f^2(x_1,x_1) = f^2(x_2,x_2)$, so that $(x_1,x_1) = (x_2,x_2)$. Hence, $x_1=x_2$.


structure_id: squaring_sets
property_id: essentially injective
type: functor
proof:
Let $X,Y$ be two sets with $X^2 \cong Y^2$. If $X$ is finite, then $X^2$ and hence $Y^2$ is finite, so that also $Y$ is finite. For the cardinalities we deduce $\card(X)^2 = \card(Y)^2$ and hence $\card(X) = \card(Y)$. This yields $X \cong Y$. If $X$ is infinite, then $Y$ is infinite, and $X \cong X^2 \cong Y^2 \cong Y$.


structure_id: squaring_sets
property_id: full on isomorphisms
type: functor
proof:
If $X$ has more than one element, the swap map $X^2 \to X^2$, $(x,y) \mapsto (y,x)$ is not induced by any map $X \to X$.


structure_id: squaring_sets
property_id: preserves coequalizers
type: functor
proof:
Consider the two maps $x,y : \{\ast\} \rightrightarrows \{x,y\}$. Their coequalizer is the unique map $\{x,y\} \to \{\ast\}$. The coequalizer of the induced maps $\{\ast\}^2 \rightrightarrows \{x,y\}^2$ is the quotient of $\{x,y\}^2$ by the equivalence relation generated by $(x,x) \sim (y,y)$. It has three elements and is therefore not isomorphic to $\{\ast\} \cong \{\ast\}^2$.


structure_id: squaring_sets
property_id: left-invertible
type: functor
proof:
Assume that there is a functor $F : \Set \to \Set$ with natural isomorphisms  $$\alpha_X : X \to F(X^2).$$ By the Yoneda Lemma, there is some $u \in F(1)$ such that $\alpha_X$ is given by $$\alpha_X(x) = F((x,x) : 1 \to X^2)(u)$$ for $x \in X$. Since $(x,x)$ factors through the diagonal $\Delta_X : X \to X^2$, and $\alpha_X$ is surjective, it follows that $F(\Delta_X) : F(X) \to F(X^2)$ is surjective. It is also injective since $\Delta_X$ is a split monomorphism. Hence, $F(\Delta_X)$ is an isomorphism. Thus, $X \cong F(X^2) \cong F(X)$. But then also $X \cong F(X^2) \cong X^2$, and we obtain a contradiction.


structure_id: torsion
property_id: preserves equalizers
type: functor
proof:
Let $E \subseteq A$ be the equalizer of two homomorphisms $f,g : A \rightrightarrows B$. If $a \in T(A)$ is equalized by $T(f),T(g) : T(A) \rightrightarrows T(B)$, then $a \in A$ is equalized by $f,g$, so that $a \in E$. It follows $a \in T(A) \cap E = T(E)$.


structure_id: torsion
property_id: finitary
type: functor
proof:
Let $(A_i)$ be a filtered diagram of abelian groups. We need to show that the canonical map $$\colim_i T(A_i) \to T(\colim_i A_i)$$ is an isomorphism.
Injectivity: Every element of $\colim_i T(A_i)$ is induced by an element in some $T(A_i)$. Assume it lies in the kernel, i.e. it becomes zero in $\colim_i A_i$. Then there exists some $i \to j$ such that its image in $A_j$ is zero. Hence its image in $T(A_j)$ is zero, and we are done.
Surjectivity: Assume we have a torsion element in $\colim_i A_i$. It has a preimage in some $A_i$. When we multiply it with some integer $\geq 1$, it becomes zero in the colimit. This means that there exists some $i \to j$ such that this multiple becomes zero in $A_j$. Therefore, the image in $A_j$ is a torsion element. It provides the desired preimage.


structure_id: torsion
property_id: dominant
type: functor
proof:
The essential image consists precisely of the abelian torsion groups, and these are closed under retracts. For example, $\IZ$ is not contained in it.


structure_id: torsion
property_id: faithful
type: functor
proof:
We have $T(\IZ) = 0$, but $\IZ \not\cong 0$. Thus, faithfulness is failing for $0, \id_{\IZ} : \IZ \rightrightarrows \IZ$.


structure_id: torsion
property_id: full
type: functor
proof:
Let $p$ be a prime. The canonical map from $\Hom(\IZ/p^2,\IZ/p)$ to $\Hom(T(\IZ/p^2),T(\IZ/p)) = \Hom(p \IZ/p^2, \IZ/p)$ is not surjective: The homomorphism $[p] \mapsto [1]$ is not contained in its image.


structure_id: torsion
property_id: preserves epimorphisms
type: functor
proof:
Let $p$ be a prime. The surjective homomorphism $\IZ \to \IZ/p$ is mapped to the homomorphism $0 \to \IZ/p$, which is not surjective.


structure_id: torsion
property_id: preserves products
type: functor
proof:
The canonical homomorphism $$\textstyle T(\prod_{n \geq 1} \IZ/n) \to \prod_{n \geq 1} T(\IZ/n) = \prod_{n \geq 1} \IZ/n$$ is not surjective since $([1],[1],\dotsc)$ is not contained in the image.


structure_id: trivial_groups
property_id: coreflector
type: functor
proof:
The constant functor $1 \to \Grp$ with value the trivial group is a left adjoint of $! : \Grp \to 1$ and fully faithful; this is because the trivial group is an initial object of $\Grp$.


structure_id: trivial_groups
property_id: reflector
type: functor
proof:
The constant functor $1 \to \Grp$ with value the trivial group is a right adjoint of $! : \Grp \to 1$ and fully faithful; this is because the trivial group is a terminal object of $\Grp$.


structure_id: trivial_sets
property_id: coreflector
type: functor
proof:
The constant functor $1 \to \Set$ with value $\varnothing$ is a left adjoint of $! : \Set \to 1$ and fully faithful; this is because $\varnothing$ is an initial object of $\Set$.


structure_id: trivial_sets
property_id: reflector
type: functor
proof:
The constant functor $1 \to \Set$ with value $\{\ast\}$ is a right adjoint of $! : \Set \to 1$ and fully faithful; this is because $\{\ast\}$ is a terminal object of $\Set$.


structure_id: trivial_sets
property_id: faithful
type: functor
proof:
Take any pair of distinct maps of sets $f,g : X \rightrightarrows Y$. They have the same image under the functor.


structure_id: trivial_sets
property_id: full
type: functor
proof:
If $X$ is a non-empty set, there is a (unique) morphism ${!}(X) \to {!}(\varnothing)$, but no morphism $X \to \varnothing$.


structure_id: terminal-map
property_id: monomorphism
type: morphism
proof:
Since $X$ has two different elements by assumption, which trivially have the same image, the map is not injective.


structure_id: Ab
property_id: additive
type: category
proof:
Since it is abelian, it is additive (by <a href="/category-implication/abelian_definition">this result</a>).


structure_id: Ab
property_id: cokernels
type: category
proof:
Since it is abelian, it has cokernels (by <a href="/category-implication/abelian_definition">this result</a>).


structure_id: Ab
property_id: conormal
type: category
proof:
Since it is abelian, it is conormal (by <a href="/category-implication/abelian_definition">this result</a>).


structure_id: Ab
property_id: kernels
type: category
proof:
Since it is abelian, it has kernels (by <a href="/category-implication/abelian_definition">this result</a>).


structure_id: Ab
property_id: normal
type: category
proof:
Since it is abelian, it is normal (by <a href="/category-implication/abelian_definition">this result</a>).


structure_id: Ab
property_id: regular
type: category
proof:
Since it is abelian, it is regular (by <a href="/category-implication/abelian_implies_regular">this result</a>).


structure_id: Ab
property_id: locally strongly finitely presentable
type: category
proof:
Since it is finitary algebraic, it is locally strongly finitely presentable (by <a href="/category-implication/finitary_algebraic_consequence">this result</a>).


structure_id: Ab
property_id: well-copowered
type: category
proof:
Since it is finitary algebraic, it is well-copowered (by <a href="/category-implication/algebraic_well-copowered">this result</a>).


structure_id: Ab
property_id: generator
type: category
proof:
Since it is finitary algebraic, it has a generator (by <a href="/category-implication/free-algebra-generates">this result</a>).


structure_id: Ab
property_id: locally essentially small
type: category
proof:
Since it is locally small, it is locally essentially small (by <a href="/category-implication/locally_small_consequence">this result</a>).


structure_id: Ab
property_id: coregular
type: category
proof:
Since it is abelian, it is coregular (by <a href="/category-implication/dual_abelian_implies_regular">this result</a>).


structure_id: Ab
property_id: locally finitely presentable
type: category
proof:
Since it is locally strongly finitely presentable, it is locally finitely presentable (by <a href="/category-implication/locally_strongly_finitely_presentable_raise">this result</a>).


structure_id: Ab
property_id: cocomplete
type: category
proof:
Since it is locally strongly finitely presentable, it is cocomplete (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>).


structure_id: Ab
property_id: generalized variety
type: category
proof:
Since it is locally strongly finitely presentable, it is a generalized variety (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>).


structure_id: Ab
property_id: finite products
type: category
proof:
Since it is additive, it has finite products (by <a href="/category-implication/additive_definition">this result</a>).


structure_id: Ab
property_id: preadditive
type: category
proof:
Since it is additive, it is preadditive (by <a href="/category-implication/additive_definition">this result</a>).


structure_id: Ab
property_id: biproducts
type: category
proof:
Since it is additive, it has biproducts (by <a href="/category-implication/additive_biproducts">this result</a>).


structure_id: Ab
property_id: multi-algebraic
type: category
proof:
Since it is locally strongly finitely presentable, it is multi-algebraic (by <a href="/category-implication/varieties_are_multi-algebraic">this result</a>).


structure_id: Ab
property_id: finitely complete
type: category
proof:
Since it is regular, it is finitely complete (by <a href="/category-implication/regular_def">this result</a>).


structure_id: Ab
property_id: zero morphisms
type: category
proof:
Since it has kernels, it has zero morphisms (by <a href="/category-implication/kernels_condition">this result</a>).


structure_id: Ab
property_id: generating set
type: category
proof:
Since it has a generator, it has a generating set (by <a href="/category-implication/generator_consequence">this result</a>).


structure_id: Ab
property_id: inhabited
type: category
proof:
Since it has a generator, it is inhabited (by <a href="/category-implication/generator_consequence">this result</a>).


structure_id: Ab
property_id: mono-regular
type: category
proof:
Since it is normal, it is mono-regular (by <a href="/category-implication/mono_regular_via_kernels">this result</a>).


structure_id: Ab
property_id: finite coproducts
type: category
proof:
Since it is additive, it has finite coproducts (by <a href="/category-implication/dual_additive_definition">this result</a>).


structure_id: Ab
property_id: finitely cocomplete
type: category
proof:
Since it is coregular, it is finitely cocomplete (by <a href="/category-implication/dual_regular_def">this result</a>).


structure_id: Ab
property_id: epi-regular
type: category
proof:
Since it is conormal, it is epi-regular (by <a href="/category-implication/dual_mono_regular_via_kernels">this result</a>).


structure_id: Ab
property_id: Malcev
type: category
proof:
Since it is additive and is finitely complete, it is Malcev (by <a href="/category-implication/malcev_additive_criterion">this result</a>).


structure_id: Ab
property_id: unital
type: category
proof:
Since it has biproducts and is finitely complete, it is unital (by <a href="/category-implication/biproducts_unital">this result</a>).


structure_id: Ab
property_id: finitely accessible
type: category
proof:
Since it is locally finitely presentable, it is finitely accessible (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>).


structure_id: Ab
property_id: exact filtered colimits
type: category
proof:
Since it is locally finitely presentable, it has exact filtered colimits (by <a href="/category-implication/locally_finitely_presentable_consequence">this result</a>).


structure_id: Ab
property_id: locally ℵ₁-presentable
type: category
proof:
Since it is locally finitely presentable, it is locally ℵ₁-presentable (by <a href="/category-implication/locally_finitely_presentable_raise">this result</a>).


structure_id: Ab
property_id: sifted colimits
type: category
proof:
Since it is a generalized variety, it has sifted colimits (by <a href="/category-implication/generalized_variety_require_sifted_colimit">this result</a>).


structure_id: Ab
property_id: ℵ₁-accessible
type: category
proof:
Since it is a generalized variety, it is ℵ₁-accessible (by <a href="/category-implication/generalized_variety_implies_accessible">this result</a>).


structure_id: Ab
property_id: locally finitely multi-presentable
type: category
proof:
Since it is multi-algebraic, it is locally finitely multi-presentable (by <a href="/category-implication/multi-algebraic_implies_locally_finitely_multi-presentable">this result</a>).


structure_id: Ab
property_id: multi-cocomplete
type: category
proof:
Since it is multi-algebraic, it is multi-cocomplete (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>).


structure_id: Ab
property_id: effective congruences
type: category
proof:
Since it is multi-algebraic, it has effective congruences (by <a href="/category-implication/multi-algebraic_implies_effective_congruences">this result</a>).


structure_id: Ab
property_id: equalizers
type: category
proof:
Since it is finitely complete, it has equalizers (by <a href="/category-implication/finitely_complete_characterization">this result</a>).


structure_id: Ab
property_id: quotients of congruences
type: category
proof:
Since it has cokernels and has kernels and is preadditive, it has quotients of congruences (by <a href="/category-implication/congruence_quotients_via_cokernels">this result</a>).


structure_id: Ab
property_id: strongly connected
type: category
proof:
Since it is inhabited and has zero morphisms, it is strongly connected (by <a href="/category-implication/zero_morphisms_mean_strongly_connected">this result</a>).


structure_id: Ab
property_id: filtered
type: category
proof:
Since it is finitely cocomplete, it is filtered (by <a href="/category-implication/filtered_criterion">this result</a>).


structure_id: Ab
property_id: balanced
type: category
proof:
Since it is mono-regular, it is balanced (by <a href="/category-implication/mono_regular_consequence">this result</a>).


structure_id: Ab
property_id: binary products
type: category
proof:
Since it has finite products, it has binary products (by <a href="/category-implication/finite_products_characterization">this result</a>).


structure_id: Ab
property_id: terminal object
type: category
proof:
Since it has finite products, it has a terminal object (by <a href="/category-implication/finite_products_characterization">this result</a>).


structure_id: Ab
property_id: finite powers
type: category
proof:
Since it has finite products, it has finite powers (by <a href="/category-implication/finite_products_include_finite_powers">this result</a>).


structure_id: Ab
property_id: co-Malcev
type: category
proof:
Since it is additive and is finitely cocomplete, it is co-Malcev (by <a href="/category-implication/dual_malcev_additive_criterion">this result</a>).


structure_id: Ab
property_id: counital
type: category
proof:
Since it has biproducts and is finitely cocomplete, it is counital (by <a href="/category-implication/dual_biproducts_unital">this result</a>).


structure_id: Ab
property_id: connected colimits
type: category
proof:
Since it is cocomplete, it has connected colimits (by <a href="/category-implication/dual_complete_consequence">this result</a>).


structure_id: Ab
property_id: coequalizers
type: category
proof:
Since it is cocomplete, it has coequalizers (by <a href="/category-implication/dual_complete_characterization">this result</a>).


structure_id: Ab
property_id: coproducts
type: category
proof:
Since it is cocomplete, it has coproducts (by <a href="/category-implication/dual_complete_characterization">this result</a>).


structure_id: Ab
property_id: well-powered
type: category
proof:
Since it is coregular and is mono-regular and is well-copowered, it is well-powered (by <a href="/category-implication/dual_regular_well-powered_well-copowered">this result</a>).


structure_id: Ab
property_id: coquotients of cocongruences
type: category
proof:
Since it has cokernels and has kernels and is preadditive, it has coquotients of cocongruences (by <a href="/category-implication/dual_congruence_quotients_via_cokernels">this result</a>).


structure_id: Ab
property_id: effective cocongruences
type: category
proof:
Since it has cokernels and is conormal and is preadditive, it has effective cocongruences (by <a href="/category-implication/dual_preadditive_kernels_normal_imply_effective_congruences">this result</a>).


structure_id: Ab
property_id: cofiltered
type: category
proof:
Since it is finitely complete, it is cofiltered (by <a href="/category-implication/dual_filtered_criterion">this result</a>).


structure_id: Ab
property_id: binary coproducts
type: category
proof:
Since it has finite coproducts, it has binary coproducts (by <a href="/category-implication/dual_finite_products_characterization">this result</a>).


structure_id: Ab
property_id: initial object
type: category
proof:
Since it has finite coproducts, it has an initial object (by <a href="/category-implication/dual_finite_products_characterization">this result</a>).


structure_id: Ab
property_id: finite copowers
type: category
proof:
Since it has finite coproducts, it has finite copowers (by <a href="/category-implication/dual_finite_products_include_finite_powers">this result</a>).


structure_id: Ab
property_id: pointed
type: category
proof:
Since it is unital, it is pointed (by <a href="/category-implication/unital_assumptions">this result</a>).


structure_id: Ab
property_id: locally presentable
type: category
proof:
Since it is locally ℵ₁-presentable, it is locally presentable (by <a href="/category-implication/locally_countably_presentable_raise">this result</a>).


structure_id: Ab
property_id: accessible
type: category
proof:
Since it is ℵ₁-accessible, it is accessible (by <a href="/category-implication/countably_accessible_special_case">this result</a>).


structure_id: Ab
property_id: ℵ₁-filtered colimits
type: category
proof:
Since it is ℵ₁-accessible, it has ℵ₁-filtered colimits (by <a href="/category-implication/countably_accessible_special_case">this result</a>).


structure_id: Ab
property_id: filtered colimits
type: category
proof:
Since it is finitely accessible, it has filtered colimits (by <a href="/category-implication/accessible_require_filtered_colimit">this result</a>).


structure_id: Ab
property_id: connected limits
type: category
proof:
Since it is locally finitely multi-presentable, it has connected limits (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>).


structure_id: Ab
property_id: filtered-colimit-stable monomorphisms
type: category
proof:
Since it is locally finitely multi-presentable, it has filtered-colimit-stable monomorphisms (by <a href="/category-implication/locally-finitely-multi-presentable_stable-monos">this result</a>).


structure_id: Ab
property_id: Grothendieck abelian
type: category
proof:
Since it is abelian and has coproducts and has exact filtered colimits and has a generator, it is Grothendieck abelian (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>).


structure_id: Ab
property_id: connected
type: category
proof:
Since it has a terminal object, it is connected (by <a href="/category-implication/multi-terminal_with_connected">this result</a>).


structure_id: Ab
property_id: multi-terminal object
type: category
proof:
Since it has a terminal object, it has a multi-terminal object (by <a href="/category-implication/multi-terminal_with_connected">this result</a>).


structure_id: Ab
property_id: Barr-exact
type: category
proof:
Since it has effective congruences and is regular, it is Barr-exact (by <a href="/category-implication/Barr-exact_definition">this result</a>).


structure_id: Ab
property_id: semi-strongly connected
type: category
proof:
Since it is strongly connected, it is semi-strongly connected (by <a href="/category-implication/strongly_connected_consequence">this result</a>).


structure_id: Ab
property_id: disjoint finite products
type: category
proof:
Since it has finite products and is strongly connected, it has disjoint finite products (by <a href="/category-implication/strongly_connected_disjoint_products">this result</a>).


structure_id: Ab
property_id: coreflexive equalizers
type: category
proof:
Since it has equalizers, it has coreflexive equalizers (by <a href="/category-implication/coreflexive_equalizers_are_equalizers">this result</a>).


structure_id: Ab
property_id: Cauchy complete
type: category
proof:
Since it has equalizers, it is Cauchy complete (by <a href="/category-implication/equalizers_consequence">this result</a>).


structure_id: Ab
property_id: cartesian filtered colimits
type: category
proof:
Since it has exact filtered colimits, it has cartesian filtered colimits (by <a href="/category-implication/exact_includes_cartesian_filtered_colimits">this result</a>).


structure_id: Ab
property_id: sifted
type: category
proof:
Since it is filtered, it is sifted (by <a href="/category-implication/filtered_is_sifted">this result</a>).


structure_id: Ab
property_id: ℵ₁-filtered
type: category
proof:
Since it has a terminal object, it is ℵ₁-filtered (by <a href="/category-implication/terminal_object_yields_kappa_filtered">this result</a>).


structure_id: Ab
property_id: reflexive coequalizers
type: category
proof:
Since it has sifted colimits, it has reflexive coequalizers (by <a href="/category-implication/sifted_colimits_consequence">this result</a>).


structure_id: Ab
property_id: binary powers
type: category
proof:
Since it has binary products, it has binary powers (by <a href="/category-implication/binary_products_include_binary_powers">this result</a>).


structure_id: Ab
property_id: pullbacks
type: category
proof:
Since it has binary products and has equalizers, it has pullbacks (by <a href="/category-implication/pullbacks_criterion">this result</a>).


structure_id: Ab
property_id: multi-initial object
type: category
proof:
Since it is multi-cocomplete, it has a multi-initial object (by <a href="/category-implication/dual_multi-terminal_special_case">this result</a>).


structure_id: Ab
property_id: Barr-coexact
type: category
proof:
Since it is coregular and has effective cocongruences, it is Barr-coexact (by <a href="/category-implication/dual_Barr-exact_definition">this result</a>).


structure_id: Ab
property_id: disjoint finite coproducts
type: category
proof:
Since it has finite coproducts and is strongly connected, it has disjoint finite coproducts (by <a href="/category-implication/dual_strongly_connected_disjoint_products">this result</a>).


structure_id: Ab
property_id: cosifted
type: category
proof:
Since it is cofiltered, it is cosifted (by <a href="/category-implication/dual_filtered_is_sifted">this result</a>).


structure_id: Ab
property_id: ℵ₁-cofiltered
type: category
proof:
Since it has an initial object, it is ℵ₁-cofiltered (by <a href="/category-implication/dual_terminal_object_yields_kappa_filtered">this result</a>).


structure_id: Ab
property_id: copowers
type: category
proof:
Since it has coproducts, it has copowers (by <a href="/category-implication/dual_products_include_powers">this result</a>).


structure_id: Ab
property_id: ℵ₂-small coproducts
type: category
proof:
Since it has coproducts, it has ℵ₂-small coproducts (by <a href="/category-implication/dual_products_include_aleph2-small_products">this result</a>).


structure_id: Ab
property_id: binary copowers
type: category
proof:
Since it has binary coproducts, it has binary copowers (by <a href="/category-implication/dual_binary_products_include_binary_powers">this result</a>).


structure_id: Ab
property_id: pushouts
type: category
proof:
Since it has binary coproducts and has coequalizers, it has pushouts (by <a href="/category-implication/dual_pullbacks_criterion">this result</a>).


structure_id: Ab
property_id: wide pushouts
type: category
proof:
Since it has connected colimits, it has wide pushouts (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>).


structure_id: Ab
property_id: complete
type: category
proof:
Since it is locally presentable, it is complete (by <a href="/category-implication/locally_presentable_another_definition">this result</a>).


structure_id: Ab
property_id: locally multi-presentable
type: category
proof:
Since it is accessible and has connected limits, it is locally multi-presentable (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>).


structure_id: Ab
property_id: cogenerator
type: category
proof:
Since it is Grothendieck abelian, it has a cogenerator (by <a href="/category-implication/grothendieck_abelian_cogenerator">this result</a>).


structure_id: Ab
property_id: disjoint coproducts
type: category
proof:
Since it has coproducts and has disjoint finite coproducts, it has disjoint coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>).


structure_id: Ab
property_id: directed colimits
type: category
proof:
Since it has filtered colimits, it has directed colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>).


structure_id: Ab
property_id: wide pullbacks
type: category
proof:
Since it has connected limits, it has wide pullbacks (by <a href="/category-implication/connected_limits_characterization">this result</a>).


structure_id: Ab
property_id: cosifted limits
type: category
proof:
Since it has connected limits, it has cosifted limits (by <a href="/category-implication/dual_sifted_categories_are_connected">this result</a>).


structure_id: Ab
property_id: countable coproducts
type: category
proof:
Since it has ℵ₂-small coproducts, it has countable coproducts (by <a href="/category-implication/dual_aleph2-small_products_consequences">this result</a>).


structure_id: Ab
property_id: ℵ₂-small copowers
type: category
proof:
Since it has ℵ₂-small coproducts, it has ℵ₂-small copowers (by <a href="/category-implication/dual_aleph2-small_products_consequences">this result</a>).


structure_id: Ab
property_id: locally poly-presentable
type: category
proof:
Since it is accessible and has wide pullbacks, it is locally poly-presentable (by <a href="/category-implication/locally_poly-presentable_definition">this result</a>).


structure_id: Ab
property_id: products
type: category
proof:
Since it is complete, it has products (by <a href="/category-implication/complete_characterization">this result</a>).


structure_id: Ab
property_id: multi-complete
type: category
proof:
Since it is complete, it is multi-complete (by <a href="/category-implication/multi-complete_generalize_limits">this result</a>).


structure_id: Ab
property_id: cofiltered limits
type: category
proof:
Since it has wide pullbacks, it has cofiltered limits (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>).


structure_id: Ab
property_id: sequential colimits
type: category
proof:
Since it has directed colimits, it has sequential colimits (by <a href="/category-implication/dual_directed_limits_consequence">this result</a>).


structure_id: Ab
property_id: cogenerating set
type: category
proof:
Since it has a cogenerator, it has a cogenerating set (by <a href="/category-implication/dual_generator_consequence">this result</a>).


structure_id: Ab
property_id: countable copowers
type: category
proof:
Since it has countable coproducts, it has countable copowers (by <a href="/category-implication/dual_countable_products_consequence">this result</a>).


structure_id: Ab
property_id: CIP
type: category
proof:
Since it has biproducts and has filtered colimits and has filtered-colimit-stable monomorphisms and has products, it satisfies CIP (by <a href="/category-implication/CIP_criterion">this result</a>).


structure_id: Ab
property_id: powers
type: category
proof:
Since it has products, it has powers (by <a href="/category-implication/products_include_powers">this result</a>).


structure_id: Ab
property_id: ℵ₂-small products
type: category
proof:
Since it has products, it has ℵ₂-small products (by <a href="/category-implication/products_include_aleph2-small_products">this result</a>).


structure_id: Ab
property_id: disjoint products
type: category
proof:
Since it has disjoint finite products and has products, it has disjoint products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>).


structure_id: Ab
property_id: cocartesian cofiltered limits
type: category
proof:
Since it has biproducts and has cofiltered limits, it has cocartesian cofiltered limits (by <a href="/category-implication/dual_biproducts_cartesian_filtered_colimits">this result</a>).


structure_id: Ab
property_id: directed limits
type: category
proof:
Since it has cofiltered limits, it has directed limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>).


structure_id: Ab
property_id: ℵ₁-cofiltered limits
type: category
proof:
Since it has cofiltered limits, it has ℵ₁-cofiltered limits (by <a href="/category-implication/dual_aleph1-filtered-colimits-include-filtered-colimits">this result</a>).


structure_id: Ab
property_id: sequential limits
type: category
proof:
Since it has directed limits, it has sequential limits (by <a href="/category-implication/directed_limits_consequence">this result</a>).


structure_id: Ab
property_id: countable products
type: category
proof:
Since it has ℵ₂-small products, it has countable products (by <a href="/category-implication/aleph2-small_products_consequences">this result</a>).


structure_id: Ab
property_id: ℵ₂-small powers
type: category
proof:
Since it has ℵ₂-small products, it has ℵ₂-small powers (by <a href="/category-implication/aleph2-small_products_consequences">this result</a>).


structure_id: Ab
property_id: countable powers
type: category
proof:
Since it has countable products, it has countable powers (by <a href="/category-implication/countable_products_consequence">this result</a>).


structure_id: Ab
property_id: trivial
type: category
proof:
Assume for contradiction that it is trivial. Then it would be split abelian (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: discrete
type: category
proof:
Assume for contradiction that it is discrete. Then it would be skeletal (by <a href="/category-implication/discrete_consequences">this result</a>) – contradiction.


structure_id: Ab
property_id: gaunt
type: category
proof:
Assume for contradiction that it is gaunt. Then it would be skeletal (by <a href="/category-implication/gaunt_characterization">this result</a>) – contradiction.


structure_id: Ab
property_id: direct
type: category
proof:
Assume for contradiction that it is direct. Then it would be skeletal (by <a href="/category-implication/direct_implies_one-way">this result</a>) – contradiction.


structure_id: Ab
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Assume for contradiction that it has cofiltered-limit-stable epimorphisms. Then it would have biproducts and would have cofiltered limits and would have cofiltered-limit-stable epimorphisms and would have coproducts, so it would satisfy CSP (by <a href="/category-implication/dual_CIP_criterion">this result</a>) – contradiction.


structure_id: Ab
property_id: inverse
type: category
proof:
Assume for contradiction that it is inverse. Then it would be skeletal (by <a href="/category-implication/dual_direct_implies_one-way">this result</a>) – contradiction.


structure_id: Ab
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would satisfy CIP and would be self-dual, so it would satisfy CSP (by <a href="/category-implication/self-dual_CIP">this result</a>) – contradiction.


structure_id: Ab
property_id: natural numbers object
type: category
proof:
Assume for contradiction that it has a natural numbers object. Then it would have a natural numbers object and would be pointed, so it would be trivial (by <a href="/category-implication/nno_pointed_case">this result</a>) – contradiction.


structure_id: Ab
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be strongly connected and would be thin, so it would be trivial (by <a href="/category-implication/thin_not_strongly_connected">this result</a>) – contradiction.


structure_id: Ab
property_id: essentially discrete
type: category
proof:
Assume for contradiction that it is essentially discrete. Then it would be connected and would be essentially discrete, so it would be trivial (by <a href="/category-implication/essentially_discrete_trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be self-dual (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: strict initial object
type: category
proof:
Assume for contradiction that it has a strict initial object. Then it would be pointed and would have a strict initial object, so it would be trivial (by <a href="/category-implication/strict_initial_trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: regular subobject classifier
type: category
proof:
Assume for contradiction that it has a regular subobject classifier. Then it would be additive and would have a regular subobject classifier, so it would be trivial (by <a href="/category-implication/additive_trivial_condition">this result</a>) – contradiction.


structure_id: Ab
property_id: regular-subobject-trivial
type: category
proof:
Assume for contradiction that it is regular-subobject-trivial. Then it would have disjoint finite coproducts and would be regular-subobject-trivial, so it would be trivial (by <a href="/category-implication/disjoint_finite_coproducts_disallows_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: regular-quotient-trivial
type: category
proof:
Assume for contradiction that it is regular-quotient-trivial. Then it would have disjoint finite coproducts and would be regular-quotient-trivial, so it would be trivial (by <a href="/category-implication/disjoint_finite_coproducts_disallows_regular-quotient-trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: exact cofiltered limits
type: category
proof:
Assume for contradiction that it has exact cofiltered limits. Then it would have cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_exact_filtered_colimits_monos">this result</a>) – contradiction.


structure_id: Ab
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would have cofiltered limits and would be right cancellative, so it would have cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_filtered_monos_trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: quotient-trivial
type: category
proof:
Assume for contradiction that it is quotient-trivial. Then it would have cofiltered limits and would be quotient-trivial, so it would have cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_filtered_monos_iso">this result</a>) – contradiction.


structure_id: Ab
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be Cauchy complete and would be essentially finite, so it would have cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_finite_filtered_colimits">this result</a>) – contradiction.


structure_id: Ab
property_id: strict terminal object
type: category
proof:
Assume for contradiction that it has a strict terminal object. Then it would be pointed and would have a strict terminal object, so it would be trivial (by <a href="/category-implication/dual_strict_initial_trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: regular quotient object classifier
type: category
proof:
Assume for contradiction that it has a regular quotient object classifier. Then it would be additive and would have a regular quotient object classifier, so it would be trivial (by <a href="/category-implication/dual_additive_trivial_condition">this result</a>) – contradiction.


structure_id: Ab
property_id: countably distributive
type: category
proof:
Assume for contradiction that it is countably distributive. Then it would have a natural numbers object (by <a href="/category-implication/nno_criterion">this result</a>) – contradiction.


structure_id: Ab
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be left cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/zero-no-mono">this result</a>) – contradiction.


structure_id: Ab
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have an initial object, so it would have a strict initial object (by <a href="/category-implication/ccc_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: distributive
type: category
proof:
Assume for contradiction that it is distributive. Then it would have a strict initial object (by <a href="/category-implication/distributive_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: extensive
type: category
proof:
Assume for contradiction that it is extensive. Then it would have a strict initial object (by <a href="/category-implication/extensive_consequences">this result</a>) – contradiction.


structure_id: Ab
property_id: finite
type: category
proof:
Assume for contradiction that it is finite. Then it would be essentially finite (by <a href="/category-implication/finite_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: subobject classifier
type: category
proof:
Assume for contradiction that it has a subobject classifier. Then it would have a regular subobject classifier (by <a href="/category-implication/regular_subobject_weaker">this result</a>) – contradiction.


structure_id: Ab
property_id: subobject-trivial
type: category
proof:
Assume for contradiction that it is subobject-trivial. Then it would be regular-subobject-trivial (by <a href="/category-implication/subobject-trivial_characterization">this result</a>) – contradiction.


structure_id: Ab
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would have binary powers and would be core-thin, so it would be thin (by <a href="/category-implication/core-thin_products_thin">this result</a>) – contradiction.


structure_id: Ab
property_id: locally finite
type: category
proof:
Assume for contradiction that it is locally finite. Then it would have countable powers and would be locally finite, so it would be thin (by <a href="/category-implication/thin_via_locally-finite">this result</a>) – contradiction.


structure_id: Ab
property_id: one-way
type: category
proof:
Assume for contradiction that it is one-way. Then it would be one-way and would have zero morphisms, so it would be thin (by <a href="/category-implication/one-way_zero">this result</a>) – contradiction.


structure_id: Ab
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would be essentially small and would have powers, so it would be thin (by <a href="/category-implication/freyd_small">this result</a>) – contradiction.


structure_id: Ab
property_id: essentially countable
type: category
proof:
Assume for contradiction that it is essentially countable. Then it would have countable powers and would be essentially countable, so it would be thin (by <a href="/category-implication/freyd_countable">this result</a>) – contradiction.


structure_id: Ab
property_id: quasitopos
type: category
proof:
Assume for contradiction that it is a quasitopos. Then it would have a regular subobject classifier (by <a href="/category-implication/quasitopos_definition">this result</a>) – contradiction.


structure_id: Ab
property_id: cocartesian coclosed
type: category
proof:
Assume for contradiction that it is cocartesian coclosed. Then it would be cocartesian coclosed and would have a terminal object, so it would have a strict terminal object (by <a href="/category-implication/dual_ccc_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: codistributive
type: category
proof:
Assume for contradiction that it is codistributive. Then it would have a strict terminal object (by <a href="/category-implication/dual_distributive_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would have a strict terminal object (by <a href="/category-implication/dual_extensive_consequences">this result</a>) – contradiction.


structure_id: Ab
property_id: quotient object classifier
type: category
proof:
Assume for contradiction that it has a quotient object classifier. Then it would have a regular quotient object classifier (by <a href="/category-implication/dual_regular_subobject_weaker">this result</a>) – contradiction.


structure_id: Ab
property_id: locally copresentable
type: category
proof:
Assume for contradiction that it is locally copresentable. Then it would be locally copresentable and would be locally presentable, so it would be essentially small (by <a href="/category-implication/locally_presentable_essentially_small">this result</a>) – contradiction.


structure_id: Ab
property_id: locally cartesian closed
type: category
proof:
Assume for contradiction that it is locally cartesian closed. Then it would be locally cartesian closed and would have a terminal object, so it would be cartesian closed (by <a href="/category-implication/locally_cartesian_closed_with_terminal_is_closed">this result</a>) – contradiction.


structure_id: Ab
property_id: infinitary distributive
type: category
proof:
Assume for contradiction that it is infinitary distributive. Then it would be countably distributive (by <a href="/category-implication/infinitary_distributive_trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: infinitary extensive
type: category
proof:
Assume for contradiction that it is infinitary extensive. Then it would be extensive (by <a href="/category-implication/infinitary_extensive_finitary">this result</a>) – contradiction.


structure_id: Ab
property_id: small
type: category
proof:
Assume for contradiction that it is small. Then it would be essentially small (by <a href="/category-implication/small_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: countable
type: category
proof:
Assume for contradiction that it is countable. Then it would be essentially countable (by <a href="/category-implication/countable_consequence">this result</a>) – contradiction.


structure_id: Ab
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would be cartesian closed (by <a href="/category-implication/topos_definition">this result</a>) – contradiction.


structure_id: Ab
property_id: pretopos
type: category
proof:
Assume for contradiction that it is a pretopos. Then it would be extensive (by <a href="/category-implication/pretopos_definition">this result</a>) – contradiction.


structure_id: Ab
property_id: locally cocartesian coclosed
type: category
proof:
Assume for contradiction that it is locally cocartesian coclosed. Then it would have an initial object and would be locally cocartesian coclosed, so it would be cocartesian coclosed (by <a href="/category-implication/dual_locally_cartesian_closed_with_terminal_is_closed">this result</a>) – contradiction.


structure_id: Ab
property_id: countably codistributive
type: category
proof:
Assume for contradiction that it is countably codistributive. Then it would be codistributive (by <a href="/category-implication/dual_countably_distributive_trivial">this result</a>) – contradiction.


structure_id: Ab
property_id: infinitary coextensive
type: category
proof:
Assume for contradiction that it is infinitary coextensive. Then it would be coextensive (by <a href="/category-implication/dual_infinitary_extensive_finitary">this result</a>) – contradiction.


structure_id: Ab
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would be an elementary topos (by <a href="/category-implication/grothendieck_topos_definition">this result</a>) – contradiction.


structure_id: Ab
property_id: coaccessible
type: category
proof:
Assume for contradiction that it is coaccessible. Then it would be coaccessible and would be complete, so it would be locally copresentable (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>) – contradiction.


structure_id: Ab
property_id: infinitary codistributive
type: category
proof:
Assume for contradiction that it is infinitary codistributive. Then it would be countably codistributive (by <a href="/category-implication/dual_infinitary_distributive_trivial">this result</a>) – contradiction.


structure_id: Sh(X,Ab)
property_id: locally presentable
type: category
proof:
Since it is Grothendieck abelian, it is locally presentable (by <a href="/category-implication/grothendieck_abelian_presentable">this result</a>).


structure_id: Sh(X,Ab)
property_id: abelian
type: category
proof:
Since it is Grothendieck abelian, it is abelian (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>).


structure_id: Sh(X,Ab)
property_id: coproducts
type: category
proof:
Since it is Grothendieck abelian, it has coproducts (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>).


structure_id: Sh(X,Ab)
property_id: exact filtered colimits
type: category
proof:
Since it is Grothendieck abelian, it has exact filtered colimits (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>).


structure_id: Sh(X,Ab)
property_id: generator
type: category
proof:
Since it is Grothendieck abelian, it has a generator (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>).


structure_id: Sh(X,Ab)
property_id: accessible
type: category
proof:
Since it is locally presentable, it is accessible (by <a href="/category-implication/locally_presentable_definition">this result</a>).


structure_id: Sh(X,Ab)
property_id: cocomplete
type: category
proof:
Since it is locally presentable, it is cocomplete (by <a href="/category-implication/locally_presentable_definition">this result</a>).


structure_id: Sh(X,Ab)
property_id: filtered colimits
type: category
proof:
Since it has exact filtered colimits, it has filtered colimits (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>).


structure_id: Sh(X,Ab)
property_id: finitely complete
type: category
proof:
Since it has exact filtered colimits, it is finitely complete (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>).


structure_id: Sh(X,Ab)
property_id: filtered-colimit-stable monomorphisms
type: category
proof:
Since it has exact filtered colimits, it has filtered-colimit-stable monomorphisms (by <a href="/category-implication/exact_filtered_colimits_monos">this result</a>).


structure_id: Sh(X,Ab)
property_id: well-powered
type: category
proof:
Since it is accessible, it is well-powered (by <a href="/category-implication/accessible_well-powered">this result</a>).


structure_id: Sh(X,Ab)
property_id: Cauchy complete
type: category
proof:
Since it is accessible, it is Cauchy complete (by <a href="/category-implication/accessible_require_Cauchy_complete">this result</a>).


structure_id: Sh(X,Ab)
property_id: connected limits
type: category
proof:
Since it is complete, it has connected limits (by <a href="/category-implication/complete_consequence">this result</a>).


structure_id: Sh(X,Ab)
property_id: equalizers
type: category
proof:
Since it is complete, it has equalizers (by <a href="/category-implication/complete_characterization">this result</a>).


structure_id: Sh(X,Ab)
property_id: ℵ₁-filtered colimits
type: category
proof:
Since it has filtered colimits, it has ℵ₁-filtered colimits (by <a href="/category-implication/aleph1-filtered-colimits-include-filtered-colimits">this result</a>).


structure_id: Sh(X,Ab)
property_id: finitely cocomplete
type: category
proof:
Since it is cocomplete, it is finitely cocomplete (by <a href="/category-implication/dual_complete_consequence">this result</a>).


structure_id: Sh(X,Ab)
property_id: multi-cocomplete
type: category
proof:
Since it is cocomplete, it is multi-cocomplete (by <a href="/category-implication/dual_multi-complete_generalize_limits">this result</a>).


structure_id: Sh(X,Ab)
property_id: multi-terminal object
type: category
proof:
Since it is multi-complete, it has a multi-terminal object (by <a href="/category-implication/multi-terminal_special_case">this result</a>).


structure_id: Sh(X,Ab)
property_id: well-copowered
type: category
proof:
Since it is epi-regular and is regular and is well-powered, it is well-copowered (by <a href="/category-implication/regular_well-powered_well-copowered">this result</a>).


structure_id: Sh(X,Ab)
property_id: effective congruences
type: category
proof:
Since it has kernels and is normal and is preadditive, it has effective congruences (by <a href="/category-implication/preadditive_kernels_normal_imply_effective_congruences">this result</a>).


structure_id: Sh(X,Ab)
property_id: ℵ₁-filtered
type: category
proof:
Since it has coequalizers and has countable coproducts, it is ℵ₁-filtered (by <a href="/category-implication/aleph1-filtered_criterion">this result</a>).


structure_id: Sh(X,Ab)
property_id: sifted colimits
type: category
proof:
Since it has connected colimits, it has sifted colimits (by <a href="/category-implication/sifted_categories_are_connected">this result</a>).


structure_id: Sh(X,Ab)
property_id: reflexive coequalizers
type: category
proof:
Since it has coequalizers, it has reflexive coequalizers (by <a href="/category-implication/dual_coreflexive_equalizers_are_equalizers">this result</a>).


structure_id: Sh(X,Ab)
property_id: sequential limits
type: category
proof:
Since it has countable products and has equalizers, it has sequential limits (by <a href="/category-implication/sequential_limits_criterion">this result</a>).


structure_id: Sh(X,Ab)
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be Grothendieck abelian and would be self-dual, so it would be trivial (by <a href="/category-implication/grothendieck_abelian_self-dual">this result</a>) – contradiction.


structure_id: Sh(X,Ab)
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be finitely complete and would be right cancellative, so it would have a regular subobject classifier (by <a href="/category-implication/regular_subobjects_trivial">this result</a>) – contradiction.


structure_id: Sh(X,Ab)
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be essentially finite and would have finite powers, so it would be thin (by <a href="/category-implication/freyd_finite">this result</a>) – contradiction.


structure_id: Sh(X,Ab)
property_id: quotient-trivial
type: category
proof:
Assume for contradiction that it is quotient-trivial. Then it would be regular-quotient-trivial (by <a href="/category-implication/dual_subobject-trivial_characterization">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: finitely complete
type: category
proof:
Since it is Malcev, it is finitely complete (by <a href="/category-implication/malcev_condition">this result</a>).


structure_id: Alg(R)
property_id: finite products
type: category
proof:
Since it has disjoint finite products, it has finite products (by <a href="/category-implication/dual_disjoint_coproducts_consequence">this result</a>).


structure_id: Alg(R)
property_id: terminal object
type: category
proof:
Since it has a strict terminal object, it has a terminal object (by <a href="/category-implication/dual_strict_initial_consequence">this result</a>).


structure_id: Alg(R)
property_id: regular
type: category
proof:
Since it is locally strongly finitely presentable, it is regular (by <a href="/category-implication/many_sorted_algebraic_implies_regular">this result</a>).


structure_id: Alg(R)
property_id: filtered
type: category
proof:
Since it is ℵ₁-filtered, it is filtered (by <a href="/category-implication/aleph1-filtered-consequence">this result</a>).


structure_id: Alg(R)
property_id: quotients of congruences
type: category
proof:
Since it has effective congruences and is regular, it has quotients of congruences (by <a href="/category-implication/regular_effective_congruences_implies_quotients">this result</a>).


structure_id: Alg(R)
property_id: finite coproducts
type: category
proof:
Since it is finitely cocomplete, it has finite coproducts (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>).


structure_id: Alg(R)
property_id: coquotients of cocongruences
type: category
proof:
Since it has coreflexive equalizers, it has coquotients of cocongruences (by <a href="/category-implication/dual_congruence_quotients_are_reflexive_coequalizers">this result</a>).


structure_id: Alg(R)
property_id: initial object
type: category
proof:
Since it is connected and has a multi-initial object, it has an initial object (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>).


structure_id: Alg(R)
property_id: pushouts
type: category
proof:
Since it has wide pushouts, it has pushouts (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>).


structure_id: Alg(R)
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would have effective cocongruences (by <a href="/category-implication/reflexive_pair_trivial">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: strongly connected
type: category
proof:
Assume for contradiction that it is strongly connected. Then it would be semi-strongly connected (by <a href="/category-implication/strongly_connected_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: extensive
type: category
proof:
Assume for contradiction that it is extensive. Then it would have cofiltered limits and would be extensive and would have a terminal object, so it would have cocartesian cofiltered limits (by <a href="/category-implication/extensive_cocartesian_cofiltered_limits">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: mono-regular
type: category
proof:
Assume for contradiction that it is mono-regular. Then it would be balanced (by <a href="/category-implication/mono_regular_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: quasitopos
type: category
proof:
Assume for contradiction that it is a quasitopos. Then it would be coregular (by <a href="/category-implication/topos_implies_coregular">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be finitely cocomplete and would be thin, so it would be co-Malcev (by <a href="/category-implication/dual_malcev_thin_condition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be additive and would be finitely cocomplete, so it would be co-Malcev (by <a href="/category-implication/dual_malcev_additive_criterion">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: coaccessible
type: category
proof:
Assume for contradiction that it is coaccessible. Then it would have a cogenerating set (by <a href="/category-implication/dual_accessible_trivial_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: abelian
type: category
proof:
Assume for contradiction that it is abelian. Then it would be coregular (by <a href="/category-implication/dual_abelian_implies_regular">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would have effective cocongruences (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: cocartesian coclosed
type: category
proof:
Assume for contradiction that it is cocartesian coclosed. Then it would be cocartesian coclosed and would have cofiltered limits, so it would have cocartesian cofiltered limits (by <a href="/category-implication/dual_ccc_cartesian_filtered_colimits">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: locally cocartesian coclosed
type: category
proof:
Assume for contradiction that it is locally cocartesian coclosed. Then it would have equalizers and would be finitely cocomplete and would be locally cocartesian coclosed, so it would be coregular (by <a href="/category-implication/dual_lcc_implies_regular">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would have effective cocongruences (by <a href="/category-implication/dual_core-hin_quotients">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: Barr-coexact
type: category
proof:
Assume for contradiction that it is Barr-coexact. Then it would be coregular (by <a href="/category-implication/dual_Barr-exact_definition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have biproducts and would have cofiltered limits, so it would have cocartesian cofiltered limits (by <a href="/category-implication/dual_biproducts_cartesian_filtered_colimits">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would be coextensive and would have finite coproducts, so it would be codistributive (by <a href="/category-implication/dual_extensive_distributivity">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: cogenerator
type: category
proof:
Assume for contradiction that it has a cogenerator. Then it would have a cogenerating set (by <a href="/category-implication/dual_generator_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: epi-regular
type: category
proof:
Assume for contradiction that it is epi-regular. Then it would be balanced (by <a href="/category-implication/dual_mono_regular_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would have a cogenerating set (by <a href="/category-implication/dual_essentially_small_consequences">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be Malcev and would be self-dual, so it would be co-Malcev (by <a href="/category-implication/self-dual_Malcev">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would have finite products and would be preadditive, so it would be additive (by <a href="/category-implication/additive_definition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would be abelian (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: split abelian
type: category
proof:
Assume for contradiction that it is split abelian. Then it would be abelian (by <a href="/category-implication/split_abelian_condition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have a strict terminal object, so it would be thin (by <a href="/category-implication/ccc_no_strict_terminal">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would be inhabited and would have zero morphisms, so it would be strongly connected (by <a href="/category-implication/zero_morphisms_mean_strongly_connected">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: trivial
type: category
proof:
Assume for contradiction that it is trivial. Then it would be essentially finite (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: essentially discrete
type: category
proof:
Assume for contradiction that it is essentially discrete. Then it would be thin (by <a href="/category-implication/essentially_discrete_characterization">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: disjoint finite coproducts
type: category
proof:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would have a strict terminal object, so it would be thin (by <a href="/category-implication/disjoint_coproducts_strict">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be left cancellative (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: normal
type: category
proof:
Assume for contradiction that it is normal. Then it would be mono-regular (by <a href="/category-implication/mono_regular_via_kernels">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: essentially countable
type: category
proof:
Assume for contradiction that it is essentially countable. Then it would be essentially small (by <a href="/category-implication/essentially_countable_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: subobject classifier
type: category
proof:
Assume for contradiction that it has a subobject classifier. Then it would be mono-regular (by <a href="/category-implication/subobject_classifier_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: regular subobject classifier
type: category
proof:
Assume for contradiction that it has a regular subobject classifier. Then it would have a regular subobject classifier and would have a strict terminal object, so it would be thin (by <a href="/category-implication/regular_subobject_classifier_strict">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: regular-subobject-trivial
type: category
proof:
Assume for contradiction that it is regular-subobject-trivial. Then it would have equalizers and would be regular-subobject-trivial, so it would be thin (by <a href="/category-implication/thin_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: subobject-trivial
type: category
proof:
Assume for contradiction that it is subobject-trivial. Then it would be mono-regular (by <a href="/category-implication/subobject-trivial_characterization">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: one-way
type: category
proof:
Assume for contradiction that it is one-way. Then it would be core-thin (by <a href="/category-implication/one-way_implies_core-thin">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would be epi-regular (by <a href="/category-implication/topos_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would have a cogenerator (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: strict initial object
type: category
proof:
Assume for contradiction that it has a strict initial object. Then it would have disjoint finite products and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_disjoint_coproducts_strict">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: conormal
type: category
proof:
Assume for contradiction that it is conormal. Then it would be epi-regular (by <a href="/category-implication/dual_mono_regular_via_kernels">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: regular-quotient-trivial
type: category
proof:
Assume for contradiction that it is regular-quotient-trivial. Then it would have coequalizers and would be regular-quotient-trivial, so it would be thin (by <a href="/category-implication/dual_thin_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: natural numbers object
type: category
proof:
Assume for contradiction that it has a natural numbers object. Then it would have a natural numbers object and would have a strict terminal object, so it would be one-way (by <a href="/category-implication/nno_terminal">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: disjoint coproducts
type: category
proof:
Assume for contradiction that it has disjoint coproducts. Then it would have disjoint finite coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: kernels
type: category
proof:
Assume for contradiction that it has kernels. Then it would have zero morphisms (by <a href="/category-implication/kernels_condition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: CIP
type: category
proof:
Assume for contradiction that it satisfies CIP. Then it would have zero morphisms (by <a href="/category-implication/CIP_assumption">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: pointed
type: category
proof:
Assume for contradiction that it is pointed. Then it would have zero morphisms (by <a href="/category-implication/pointed_characterization">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: cokernels
type: category
proof:
Assume for contradiction that it has cokernels. Then it would have zero morphisms (by <a href="/category-implication/dual_kernels_condition">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: CSP
type: category
proof:
Assume for contradiction that it satisfies CSP. Then it would have zero morphisms (by <a href="/category-implication/dual_CIP_assumption">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: unital
type: category
proof:
Assume for contradiction that it is unital. Then it would be pointed (by <a href="/category-implication/unital_assumptions">this result</a>) – contradiction.


structure_id: Alg(R)
property_id: counital
type: category
proof:
Assume for contradiction that it is counital. Then it would be pointed (by <a href="/category-implication/dual_unital_assumptions">this result</a>) – contradiction.


structure_id: Ban
property_id: cocomplete
type: category
proof:
Since it is locally ℵ₁-presentable, it is cocomplete (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>).


structure_id: Ban
property_id: ℵ₁-accessible
type: category
proof:
Since it is locally ℵ₁-presentable, it is ℵ₁-accessible (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>).


structure_id: Ban
property_id: filtered colimits
type: category
proof:
Since it has cartesian filtered colimits, it has filtered colimits (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>).


structure_id: Ban
property_id: finite products
type: category
proof:
Since it has cartesian filtered colimits, it has finite products (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>).


structure_id: Ban
property_id: coproducts
type: category
proof:
Since it satisfies CIP, it has coproducts (by <a href="/category-implication/CIP_assumption">this result</a>).


structure_id: Ban
property_id: products
type: category
proof:
Since it satisfies CIP, it has products (by <a href="/category-implication/CIP_assumption">this result</a>).


structure_id: Ban
property_id: zero morphisms
type: category
proof:
Since it satisfies CIP, it has zero morphisms (by <a href="/category-implication/CIP_assumption">this result</a>).


structure_id: Ban
property_id: initial object
type: category
proof:
Since it is pointed, it has an initial object (by <a href="/category-implication/pointed_characterization">this result</a>).


structure_id: Ban
property_id: cofiltered limits
type: category
proof:
Since it has cocartesian cofiltered limits, it has cofiltered limits (by <a href="/category-implication/dual_cartesian_filtered_colimits_condition">this result</a>).


structure_id: Ban
property_id: finite coproducts
type: category
proof:
Since it has cocartesian cofiltered limits, it has finite coproducts (by <a href="/category-implication/dual_cartesian_filtered_colimits_condition">this result</a>).


structure_id: Ban
property_id: inhabited
type: category
proof:
Since it has a cogenerator, it is inhabited (by <a href="/category-implication/dual_generator_consequence">this result</a>).


structure_id: Ban
property_id: terminal object
type: category
proof:
Since it is pointed, it has a terminal object (by <a href="/category-implication/dual_pointed_characterization">this result</a>).


structure_id: Ban
property_id: multi-initial object
type: category
proof:
Since it has an initial object, it has a multi-initial object (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>).


structure_id: Ban
property_id: generating set
type: category
proof:
Since it is accessible, it has a generating set (by <a href="/category-implication/accessible_trivial_consequence">this result</a>).


structure_id: Ban
property_id: locally multi-presentable
type: category
proof:
Since it is accessible and is multi-cocomplete, it is locally multi-presentable (by <a href="/category-implication/locally_multi-presentable_another_definition">this result</a>).


structure_id: Ban
property_id: kernels
type: category
proof:
Since it has equalizers and has zero morphisms, it has kernels (by <a href="/category-implication/kernels_criterion">this result</a>).


structure_id: Ban
property_id: cokernels
type: category
proof:
Since it has coequalizers and has zero morphisms, it has cokernels (by <a href="/category-implication/dual_kernels_criterion">this result</a>).


structure_id: Ban
property_id: well-copowered
type: category
proof:
Since it is accessible and has pushouts, it is well-copowered (by <a href="/category-implication/accessible_well-copowered">this result</a>).


structure_id: Ban
property_id: generator
type: category
proof:
Since it has coproducts and has a generating set and has zero morphisms, it has a generator (by <a href="/category-implication/generator_via_coproduct">this result</a>).


structure_id: Ban
property_id: Malcev
type: category
proof:
Assume for contradiction that it is Malcev. Then it would be Malcev and would be pointed, so it would be unital (by <a href="/category-implication/malcev_implies_unital">this result</a>) – contradiction.


structure_id: Ban
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have biproducts and would be finitely complete, so it would be unital (by <a href="/category-implication/biproducts_unital">this result</a>) – contradiction.


structure_id: Ban
property_id: locally finitely multi-presentable
type: category
proof:
Assume for contradiction that it is locally finitely multi-presentable. Then it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/locally-finitely-multi-presentable_stable-monos">this result</a>) – contradiction.


structure_id: Ban
property_id: exact filtered colimits
type: category
proof:
Assume for contradiction that it has exact filtered colimits. Then it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/exact_filtered_colimits_monos">this result</a>) – contradiction.


structure_id: Ban
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would have filtered colimits and would be left cancellative, so it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/filtered_monos_trivial">this result</a>) – contradiction.


structure_id: Ban
property_id: subobject-trivial
type: category
proof:
Assume for contradiction that it is subobject-trivial. Then it would have filtered colimits and would be subobject-trivial, so it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/filtered_monos_iso">this result</a>) – contradiction.


structure_id: Ban
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be Cauchy complete and would be essentially finite, so it would have filtered-colimit-stable monomorphisms (by <a href="/category-implication/finite_filtered_colimits">this result</a>) – contradiction.


structure_id: Ban
property_id: co-Malcev
type: category
proof:
Assume for contradiction that it is co-Malcev. Then it would be co-Malcev and would be pointed, so it would be counital (by <a href="/category-implication/dual_malcev_implies_unital">this result</a>) – contradiction.


structure_id: Ban
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be finitely complete and would be thin, so it would be Malcev (by <a href="/category-implication/malcev_thin_condition">this result</a>) – contradiction.


structure_id: Ban
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be additive and would be finitely complete, so it would be Malcev (by <a href="/category-implication/malcev_additive_criterion">this result</a>) – contradiction.


structure_id: Ban
property_id: locally finitely presentable
type: category
proof:
Assume for contradiction that it is locally finitely presentable. Then it would have exact filtered colimits (by <a href="/category-implication/locally_finitely_presentable_consequence">this result</a>) – contradiction.


structure_id: Ban
property_id: finitely accessible
type: category
proof:
Assume for contradiction that it is finitely accessible. Then it would have connected limits and would be finitely accessible, so it would be locally finitely multi-presentable (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>) – contradiction.


structure_id: Ban
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would have exact filtered colimits (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>) – contradiction.


structure_id: Ban
property_id: multi-algebraic
type: category
proof:
Assume for contradiction that it is multi-algebraic. Then it would be locally finitely multi-presentable (by <a href="/category-implication/multi-algebraic_implies_locally_finitely_multi-presentable">this result</a>) – contradiction.


structure_id: Ban
property_id: extensive
type: category
proof:
Assume for contradiction that it is extensive. Then it would have effective congruences and would be extensive, so it would be mono-regular (by <a href="/category-implication/pretopos_balanced">this result</a>) – contradiction.


structure_id: Ban
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would have a subobject classifier (by <a href="/category-implication/topos_definition">this result</a>) – contradiction.


structure_id: Ban
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would have exact filtered colimits (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>) – contradiction.


structure_id: Ban
property_id: locally strongly finitely presentable
type: category
proof:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be locally finitely presentable (by <a href="/category-implication/locally_strongly_finitely_presentable_raise">this result</a>) – contradiction.


structure_id: Ban
property_id: abelian
type: category
proof:
Assume for contradiction that it is abelian. Then it would be additive (by <a href="/category-implication/abelian_definition">this result</a>) – contradiction.


structure_id: Ban
property_id: generalized variety
type: category
proof:
Assume for contradiction that it is a generalized variety. Then it would be a generalized variety and would be multi-cocomplete, so it would be multi-algebraic (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>) – contradiction.


structure_id: Ban
property_id: essentially discrete
type: category
proof:
Assume for contradiction that it is essentially discrete. Then it would be a groupoid (by <a href="/category-implication/essentially_discrete_characterization">this result</a>) – contradiction.


structure_id: Ban
property_id: strict terminal object
type: category
proof:
Assume for contradiction that it has a strict terminal object. Then it would have disjoint finite coproducts and would have a strict terminal object, so it would be thin (by <a href="/category-implication/disjoint_coproducts_strict">this result</a>) – contradiction.


structure_id: Ban
property_id: locally cartesian closed
type: category
proof:
Assume for contradiction that it is locally cartesian closed. Then it would have disjoint finite coproducts and would be locally cartesian closed, so it would be extensive (by <a href="/category-implication/lcc_implies_extensive">this result</a>) – contradiction.


structure_id: Ban
property_id: finitary algebraic
type: category
proof:
Assume for contradiction that it is finitary algebraic. Then it would be locally strongly finitely presentable (by <a href="/category-implication/finitary_algebraic_consequence">this result</a>) – contradiction.


structure_id: Sp
property_id: generating set
type: category
proof:
Since it is essentially small, it has a generating set (by <a href="/category-implication/essentially_small_consequences">this result</a>).


structure_id: Sp
property_id: locally essentially small
type: category
proof:
Since it is essentially small, it is locally essentially small (by <a href="/category-implication/essentially_small_consequences">this result</a>).


structure_id: Sp
property_id: well-copowered
type: category
proof:
Since it is essentially small, it is well-copowered (by <a href="/category-implication/essentially_small_consequences">this result</a>).


structure_id: Sp
property_id: well-powered
type: category
proof:
Since it is essentially small, it is well-powered (by <a href="/category-implication/essentially_small_consequences">this result</a>).


structure_id: Sp
property_id: cartesian closed
type: category
proof:
Since it is an elementary topos, it is cartesian closed (by <a href="/category-implication/topos_definition">this result</a>).


structure_id: Sp
property_id: finitely complete
type: category
proof:
Since it is an elementary topos, it is finitely complete (by <a href="/category-implication/topos_definition">this result</a>).


structure_id: Sp
property_id: subobject classifier
type: category
proof:
Since it is an elementary topos, it has a subobject classifier (by <a href="/category-implication/topos_definition">this result</a>).


structure_id: Sp
property_id: disjoint finite coproducts
type: category
proof:
Since it is an elementary topos, it has disjoint finite coproducts (by <a href="/category-implication/topos_consequence">this result</a>).


structure_id: Sp
property_id: effective congruences
type: category
proof:
Since it is an elementary topos, it has effective congruences (by <a href="/category-implication/topos_consequence">this result</a>).


structure_id: Sp
property_id: epi-regular
type: category
proof:
Since it is an elementary topos, it is epi-regular (by <a href="/category-implication/topos_consequence">this result</a>).


structure_id: Sp
property_id: finitely cocomplete
type: category
proof:
Since it is an elementary topos, it is finitely cocomplete (by <a href="/category-implication/topos_consequence">this result</a>).


structure_id: Sp
property_id: locally cartesian closed
type: category
proof:
Since it is an elementary topos, it is locally cartesian closed (by <a href="/category-implication/topos_is_locally_cartesian_closed">this result</a>).


structure_id: Sp
property_id: Cauchy complete
type: category
proof:
Since it has ℵ₁-cofiltered limits, it is Cauchy complete (by <a href="/category-implication/dual_kappa-filtered-colimits_require_Cauchy_complete">this result</a>).


structure_id: Sp
property_id: finite products
type: category
proof:
Since it is cartesian closed, it has finite products (by <a href="/category-implication/ccc_condition">this result</a>).


structure_id: Sp
property_id: pullbacks
type: category
proof:
Since it is locally cartesian closed, it has pullbacks (by <a href="/category-implication/pullbacks_are_local_products">this result</a>).


structure_id: Sp
property_id: finite coproducts
type: category
proof:
Since it has disjoint finite coproducts, it has finite coproducts (by <a href="/category-implication/disjoint_coproducts_consequence">this result</a>).


structure_id: Sp
property_id: extensive
type: category
proof:
Since it has disjoint finite coproducts and is locally cartesian closed, it is extensive (by <a href="/category-implication/lcc_implies_extensive">this result</a>).


structure_id: Sp
property_id: mono-regular
type: category
proof:
Since it has a subobject classifier, it is mono-regular (by <a href="/category-implication/subobject_classifier_consequence">this result</a>).


structure_id: Sp
property_id: regular subobject classifier
type: category
proof:
Since it has a subobject classifier, it has a regular subobject classifier (by <a href="/category-implication/regular_subobject_weaker">this result</a>).


structure_id: Sp
property_id: coaccessible
type: category
proof:
Since it is Cauchy complete and is essentially small, it is coaccessible (by <a href="/category-implication/dual_small_accessible_characterization">this result</a>).


structure_id: Sp
property_id: coequalizers
type: category
proof:
Since it is finitely cocomplete, it has coequalizers (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>).


structure_id: Sp
property_id: balanced
type: category
proof:
Since it is epi-regular, it is balanced (by <a href="/category-implication/dual_mono_regular_consequence">this result</a>).


structure_id: Sp
property_id: regular
type: category
proof:
Since it has coequalizers and is finitely complete and is locally cartesian closed, it is regular (by <a href="/category-implication/lcc_implies_regular">this result</a>).


structure_id: Sp
property_id: distributive
type: category
proof:
Since it is cartesian closed and has finite coproducts, it is distributive (by <a href="/category-implication/distributive_criterion">this result</a>).


structure_id: Sp
property_id: strict initial object
type: category
proof:
Since it is extensive, it has a strict initial object (by <a href="/category-implication/extensive_consequences">this result</a>).


structure_id: Sp
property_id: quasitopos
type: category
proof:
Since it is finitely cocomplete and is finitely complete and is locally cartesian closed and has a regular subobject classifier, it is a quasitopos (by <a href="/category-implication/quasitopos_definition">this result</a>).


structure_id: Sp
property_id: quotients of congruences
type: category
proof:
Since it has reflexive coequalizers, it has quotients of congruences (by <a href="/category-implication/congruence_quotients_are_reflexive_coequalizers">this result</a>).


structure_id: Sp
property_id: co-Malcev
type: category
proof:
Since it is epi-regular and is extensive and is regular, it is co-Malcev (by <a href="/category-implication/regular_epi-regular_extensive_consequences">this result</a>).


structure_id: Sp
property_id: effective cocongruences
type: category
proof:
Since it is epi-regular and is extensive and is regular, it has effective cocongruences (by <a href="/category-implication/regular_epi-regular_extensive_consequences">this result</a>).


structure_id: Sp
property_id: coregular
type: category
proof:
Since it is a quasitopos, it is coregular (by <a href="/category-implication/topos_implies_coregular">this result</a>).


structure_id: Sp
property_id: pretopos
type: category
proof:
Since it is Barr-exact and is extensive, it is a pretopos (by <a href="/category-implication/pretopos_definition">this result</a>).


structure_id: Sp
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have finite products and would be thin, so it would have a natural numbers object (by <a href="/category-implication/nno_thin">this result</a>) – contradiction.


structure_id: Sp
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would have a generator (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: discrete
type: category
proof:
Assume for contradiction that it is discrete. Then it would be locally small (by <a href="/category-implication/discrete_consequences">this result</a>) – contradiction.


structure_id: Sp
property_id: finitary algebraic
type: category
proof:
Assume for contradiction that it is finitary algebraic. Then it would have a generator (by <a href="/category-implication/free-algebra-generates">this result</a>) – contradiction.


structure_id: Sp
property_id: small
type: category
proof:
Assume for contradiction that it is small. Then it would be locally small (by <a href="/category-implication/small_consequence">this result</a>) – contradiction.


structure_id: Sp
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be essentially countable (by <a href="/category-implication/essentially_finite_raise">this result</a>) – contradiction.


structure_id: Sp
property_id: disjoint products
type: category
proof:
Assume for contradiction that it has disjoint products. Then it would have disjoint products and would have a generating set, so it would have a generator (by <a href="/category-implication/dual_disjoint_coproduct_cogenerator">this result</a>) – contradiction.


structure_id: Sp
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have a cogenerator and would be self-dual, so it would have a generator (by <a href="/category-implication/self-dual_cogenerator">this result</a>) – contradiction.


structure_id: Sp
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would have binary copowers and would be left cancellative, so it would be thin (by <a href="/category-implication/codiagonal-no-mono">this result</a>) – contradiction.


structure_id: Sp
property_id: strict terminal object
type: category
proof:
Assume for contradiction that it has a strict terminal object. Then it would be cartesian closed and would have a strict terminal object, so it would be thin (by <a href="/category-implication/ccc_no_strict_terminal">this result</a>) – contradiction.


structure_id: Sp
property_id: countable coproducts
type: category
proof:
Assume for contradiction that it has countable coproducts. Then it would be cartesian closed and would have countable coproducts, so it would be countably distributive (by <a href="/category-implication/countably_distributive_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: finite
type: category
proof:
Assume for contradiction that it is finite. Then it would be countable (by <a href="/category-implication/finite_consequence">this result</a>) – contradiction.


structure_id: Sp
property_id: powers
type: category
proof:
Assume for contradiction that it has powers. Then it would be essentially small and would have powers, so it would be thin (by <a href="/category-implication/freyd_small">this result</a>) – contradiction.


structure_id: Sp
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would have binary powers and would be right cancellative, so it would be thin (by <a href="/category-implication/dual_codiagonal-no-mono">this result</a>) – contradiction.


structure_id: Sp
property_id: cocartesian coclosed
type: category
proof:
Assume for contradiction that it is cocartesian coclosed. Then it would be cocartesian coclosed and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_ccc_no_strict_terminal">this result</a>) – contradiction.


structure_id: Sp
property_id: disjoint finite products
type: category
proof:
Assume for contradiction that it has disjoint finite products. Then it would have disjoint finite products and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_disjoint_coproducts_strict">this result</a>) – contradiction.


structure_id: Sp
property_id: regular quotient object classifier
type: category
proof:
Assume for contradiction that it has a regular quotient object classifier. Then it would have a regular quotient object classifier and would have a strict initial object, so it would be thin (by <a href="/category-implication/dual_regular_subobject_classifier_strict">this result</a>) – contradiction.


structure_id: Sp
property_id: copowers
type: category
proof:
Assume for contradiction that it has copowers. Then it would have copowers and would be essentially small, so it would be thin (by <a href="/category-implication/dual_freyd_small">this result</a>) – contradiction.


structure_id: Sp
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would have zero morphisms (by <a href="/category-implication/preadditive_consequence">this result</a>) – contradiction.


structure_id: Sp
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have zero morphisms (by <a href="/category-implication/biproducts_definition_consequences">this result</a>) – contradiction.


structure_id: Sp
property_id: coproducts
type: category
proof:
Assume for contradiction that it has coproducts. Then it would be cartesian closed and would have coproducts, so it would be infinitary distributive (by <a href="/category-implication/infinitary_distributive_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: infinitary extensive
type: category
proof:
Assume for contradiction that it is infinitary extensive. Then it would have finite products and would be infinitary extensive, so it would be infinitary distributive (by <a href="/category-implication/infinitary_extensive_distributivity">this result</a>) – contradiction.


structure_id: Sp
property_id: normal
type: category
proof:
Assume for contradiction that it is normal. Then it would have zero morphisms (by <a href="/category-implication/normal_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: products
type: category
proof:
Assume for contradiction that it has products. Then it would have powers (by <a href="/category-implication/products_include_powers">this result</a>) – contradiction.


structure_id: Sp
property_id: filtered colimits
type: category
proof:
Assume for contradiction that it has filtered colimits. Then it would have filtered colimits and would have finite copowers, so it would have copowers (by <a href="/category-implication/copowers_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: countable copowers
type: category
proof:
Assume for contradiction that it has countable copowers. Then it would have countable copowers and would have ℵ₁-filtered colimits, so it would have copowers (by <a href="/category-implication/copowers_criterion_countable">this result</a>) – contradiction.


structure_id: Sp
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be additive and would have a regular subobject classifier, so it would be trivial (by <a href="/category-implication/additive_trivial_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: Malcev
type: category
proof:
Assume for contradiction that it is Malcev. Then it would be Malcev and would have a regular subobject classifier, so it would be regular-subobject-trivial (by <a href="/category-implication/regular_subobject_classifier_disallows_malcev">this result</a>) – contradiction.


structure_id: Sp
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would have disjoint finite products (by <a href="/category-implication/dual_extensive_consequences">this result</a>) – contradiction.


structure_id: Sp
property_id: conormal
type: category
proof:
Assume for contradiction that it is conormal. Then it would have zero morphisms (by <a href="/category-implication/dual_normal_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: ℵ₂-small coproducts
type: category
proof:
Assume for contradiction that it has ℵ₂-small coproducts. Then it would have countable coproducts (by <a href="/category-implication/dual_aleph2-small_products_consequences">this result</a>) – contradiction.


structure_id: Sp
property_id: sequential colimits
type: category
proof:
Assume for contradiction that it has sequential colimits. Then it would have finite coproducts and would have sequential colimits, so it would have countable coproducts (by <a href="/category-implication/dual_countable_products_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: cofiltered limits
type: category
proof:
Assume for contradiction that it has cofiltered limits. Then it would have cofiltered limits and would have finite powers, so it would have powers (by <a href="/category-implication/dual_copowers_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: countable powers
type: category
proof:
Assume for contradiction that it has countable powers. Then it would have countable powers and would have ℵ₁-cofiltered limits, so it would have powers (by <a href="/category-implication/dual_copowers_criterion_countable">this result</a>) – contradiction.


structure_id: Sp
property_id: finitely accessible
type: category
proof:
Assume for contradiction that it is finitely accessible. Then it would have filtered colimits (by <a href="/category-implication/accessible_require_filtered_colimit">this result</a>) – contradiction.


structure_id: Sp
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would have products (by <a href="/category-implication/complete_characterization">this result</a>) – contradiction.


structure_id: Sp
property_id: disjoint coproducts
type: category
proof:
Assume for contradiction that it has disjoint coproducts. Then it would have coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: Sp
property_id: exact filtered colimits
type: category
proof:
Assume for contradiction that it has exact filtered colimits. Then it would have filtered colimits (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: cartesian filtered colimits
type: category
proof:
Assume for contradiction that it has cartesian filtered colimits. Then it would have filtered colimits (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: filtered-colimit-stable monomorphisms
type: category
proof:
Assume for contradiction that it has filtered-colimit-stable monomorphisms. Then it would have filtered colimits (by <a href="/category-implication/filtered_monos_assumption">this result</a>) – contradiction.


structure_id: Sp
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would be cocomplete and would be extensive and would be locally cartesian closed, so it would be infinitary extensive (by <a href="/category-implication/lcc_extensive_yields_infinitary">this result</a>) – contradiction.


structure_id: Sp
property_id: directed colimits
type: category
proof:
Assume for contradiction that it has directed colimits. Then it would have filtered colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>) – contradiction.


structure_id: Sp
property_id: sifted colimits
type: category
proof:
Assume for contradiction that it has sifted colimits. Then it would have filtered colimits (by <a href="/category-implication/sifted_colimits_consequence">this result</a>) – contradiction.


structure_id: Sp
property_id: countable products
type: category
proof:
Assume for contradiction that it has countable products. Then it would have countable products and would have ℵ₁-cofiltered limits, so it would have products (by <a href="/category-implication/products_criterion_countable">this result</a>) – contradiction.


structure_id: Sp
property_id: ℵ₂-small powers
type: category
proof:
Assume for contradiction that it has ℵ₂-small powers. Then it would have countable powers (by <a href="/category-implication/aleph2-small-powers_include_countable_powers">this result</a>) – contradiction.


structure_id: Sp
property_id: sequential limits
type: category
proof:
Assume for contradiction that it has sequential limits. Then it would have finite powers and would have sequential limits, so it would have countable powers (by <a href="/category-implication/countable_powers_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: wide pullbacks
type: category
proof:
Assume for contradiction that it has wide pullbacks. Then it would have cofiltered limits (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would have coproducts (by <a href="/category-implication/grothendieck_topos_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: infinitary codistributive
type: category
proof:
Assume for contradiction that it is infinitary codistributive. Then it would have products (by <a href="/category-implication/dual_infinitary_distributive_assumption">this result</a>) – contradiction.


structure_id: Sp
property_id: exact cofiltered limits
type: category
proof:
Assume for contradiction that it has exact cofiltered limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_exact_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: cocartesian cofiltered limits
type: category
proof:
Assume for contradiction that it has cocartesian cofiltered limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_cartesian_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Sp
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Assume for contradiction that it has cofiltered-limit-stable epimorphisms. Then it would have cofiltered limits (by <a href="/category-implication/dual_filtered_monos_assumption">this result</a>) – contradiction.


structure_id: Sp
property_id: infinitary coextensive
type: category
proof:
Assume for contradiction that it is infinitary coextensive. Then it would have products (by <a href="/category-implication/dual_infinitary_extensive_assumption">this result</a>) – contradiction.


structure_id: Sp
property_id: directed limits
type: category
proof:
Assume for contradiction that it has directed limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>) – contradiction.


structure_id: Sp
property_id: cosifted limits
type: category
proof:
Assume for contradiction that it has cosifted limits. Then it would have cofiltered limits (by <a href="/category-implication/dual_sifted_colimits_consequence">this result</a>) – contradiction.


structure_id: Sp
property_id: ℵ₂-small copowers
type: category
proof:
Assume for contradiction that it has ℵ₂-small copowers. Then it would have countable copowers (by <a href="/category-implication/dual_aleph2-small-powers_include_countable_powers">this result</a>) – contradiction.


structure_id: Sp
property_id: wide pushouts
type: category
proof:
Assume for contradiction that it has wide pushouts. Then it would have filtered colimits (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>) – contradiction.


structure_id: Sp
property_id: locally finitely presentable
type: category
proof:
Assume for contradiction that it is locally finitely presentable. Then it would be cocomplete (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


structure_id: Sp
property_id: locally ℵ₁-presentable
type: category
proof:
Assume for contradiction that it is locally ℵ₁-presentable. Then it would be cocomplete (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>) – contradiction.


structure_id: Sp
property_id: locally presentable
type: category
proof:
Assume for contradiction that it is locally presentable. Then it would be cocomplete (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: locally strongly finitely presentable
type: category
proof:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be cocomplete (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: locally finitely multi-presentable
type: category
proof:
Assume for contradiction that it is locally finitely multi-presentable. Then it would be finitely accessible (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: locally poly-presentable
type: category
proof:
Assume for contradiction that it is locally poly-presentable. Then it would have wide pullbacks (by <a href="/category-implication/locally_poly-presentable_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: generalized variety
type: category
proof:
Assume for contradiction that it is a generalized variety. Then it would have sifted colimits (by <a href="/category-implication/generalized_variety_require_sifted_colimit">this result</a>) – contradiction.


structure_id: Sp
property_id: multi-complete
type: category
proof:
Assume for contradiction that it is multi-complete. Then it would have an initial object and would be multi-complete, so it would be complete (by <a href="/category-implication/multi-complete_with_initial_object">this result</a>) – contradiction.


structure_id: Sp
property_id: connected colimits
type: category
proof:
Assume for contradiction that it has connected colimits. Then it would have sifted colimits (by <a href="/category-implication/sifted_categories_are_connected">this result</a>) – contradiction.


structure_id: Sp
property_id: ℵ₂-small products
type: category
proof:
Assume for contradiction that it has ℵ₂-small products. Then it would have countable products (by <a href="/category-implication/aleph2-small_products_consequences">this result</a>) – contradiction.


structure_id: Sp
property_id: connected limits
type: category
proof:
Assume for contradiction that it has connected limits. Then it would have wide pullbacks (by <a href="/category-implication/connected_limits_characterization">this result</a>) – contradiction.


structure_id: Sp
property_id: locally copresentable
type: category
proof:
Assume for contradiction that it is locally copresentable. Then it would be complete (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>) – contradiction.


structure_id: Sp
property_id: multi-cocomplete
type: category
proof:
Assume for contradiction that it is multi-cocomplete. Then it would be multi-cocomplete and would have a terminal object, so it would be cocomplete (by <a href="/category-implication/dual_multi-complete_with_initial_object">this result</a>) – contradiction.


structure_id: Sp
property_id: locally multi-presentable
type: category
proof:
Assume for contradiction that it is locally multi-presentable. Then it would have connected limits (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>) – contradiction.


structure_id: CAlg(R)
property_id: finite products
type: category
proof:
Since it is coextensive, it has finite products (by <a href="/category-implication/dual_extensive_assumption">this result</a>).


structure_id: CAlg(R)
property_id: disjoint finite products
type: category
proof:
Since it is coextensive, it has disjoint finite products (by <a href="/category-implication/dual_extensive_consequences">this result</a>).


structure_id: CAlg(R)
property_id: codistributive
type: category
proof:
Since it is coextensive and has finite coproducts, it is codistributive (by <a href="/category-implication/dual_extensive_distributivity">this result</a>).


structure_id: CAlg(R)
property_id: cocartesian coclosed
type: category
proof:
Assume for contradiction that it is cocartesian coclosed. Then it would be cocartesian coclosed and would have countable products, so it would be countably codistributive (by <a href="/category-implication/dual_countably_distributive_criterion">this result</a>) – contradiction.


structure_id: CAlg(R)
property_id: cocartesian cofiltered limits
type: category
proof:
Assume for contradiction that it has cocartesian cofiltered limits. Then it would have cocartesian cofiltered limits and would be codistributive and would have countable products, so it would be countably codistributive (by <a href="/category-implication/dual_countably_distributive_filtered_criterion">this result</a>) – contradiction.


structure_id: CAlg(R)
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be finitely cocomplete and would be left cancellative, so it would have a regular quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_trivial">this result</a>) – contradiction.


structure_id: CAlg(R)
property_id: effective cocongruences
type: category
proof:
Assume for contradiction that it has effective cocongruences. Then it would be coextensive and would have effective cocongruences, so it would be epi-regular (by <a href="/category-implication/dual_pretopos_balanced">this result</a>) – contradiction.


structure_id: CAlg(R)
property_id: infinitary coextensive
type: category
proof:
Assume for contradiction that it is infinitary coextensive. Then it would have finite coproducts and would be infinitary coextensive, so it would be infinitary codistributive (by <a href="/category-implication/dual_infinitary_extensive_distributivity">this result</a>) – contradiction.


structure_id: CAlg(R)
property_id: countably distributive
type: category
proof:
Assume for contradiction that it is countably distributive. Then it would be distributive (by <a href="/category-implication/countably_distributive_trivial">this result</a>) – contradiction.


structure_id: CMon
property_id: finite coproducts
type: category
proof:
Since it has biproducts, it has finite coproducts (by <a href="/category-implication/biproducts_definition_consequences">this result</a>).


structure_id: CMon
property_id: finite products
type: category
proof:
Since it has biproducts, it has finite products (by <a href="/category-implication/biproducts_definition_consequences">this result</a>).


structure_id: CMon
property_id: zero morphisms
type: category
proof:
Since it has biproducts, it has zero morphisms (by <a href="/category-implication/biproducts_definition_consequences">this result</a>).


structure_id: CMon
property_id: sifted
type: category
proof:
Since it has binary coproducts and is inhabited, it is sifted (by <a href="/category-implication/coproducts_implies_sifted">this result</a>).


structure_id: CMon
property_id: cosifted
type: category
proof:
Since it has binary products and is inhabited, it is cosifted (by <a href="/category-implication/dual_coproducts_implies_sifted">this result</a>).


structure_id: CMon
property_id: cofiltered
type: category
proof:
Since it is ℵ₁-cofiltered, it is cofiltered (by <a href="/category-implication/dual_aleph1-filtered-consequence">this result</a>).


structure_id: CMon
property_id: Cauchy complete
type: category
proof:
Since it has coequalizers, it is Cauchy complete (by <a href="/category-implication/dual_equalizers_consequence">this result</a>).


structure_id: CMon
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would have a cogenerator (by <a href="/category-implication/grothendieck_abelian_cogenerator">this result</a>) – contradiction.


structure_id: CMon
property_id: cogenerating set
type: category
proof:
Assume for contradiction that it has a cogenerating set. Then it would have a cogenerating set and would have disjoint coproducts, so it would have a cogenerator (by <a href="/category-implication/disjoint_coproduct_cogenerator">this result</a>) – contradiction.


structure_id: CMon
property_id: trivial
type: category
proof:
Assume for contradiction that it is trivial. Then it would be a Grothendieck topos (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


structure_id: CompHaus
property_id: complete
type: category
proof:
Since it has equalizers and has products, it is complete (by <a href="/category-implication/complete_characterization">this result</a>).


structure_id: CompHaus
property_id: effective congruences
type: category
proof:
Since it is Barr-exact, it has effective congruences (by <a href="/category-implication/Barr-exact_definition">this result</a>).


structure_id: CompHaus
property_id: regular
type: category
proof:
Since it is Barr-exact, it is regular (by <a href="/category-implication/Barr-exact_definition">this result</a>).


structure_id: CompHaus
property_id: connected
type: category
proof:
Since it is semi-strongly connected, it is connected (by <a href="/category-implication/semi-strongly_connected_consequence">this result</a>).


structure_id: CompHaus
property_id: finite coproducts
type: category
proof:
Since it is extensive, it has finite coproducts (by <a href="/category-implication/extensive_assumption">this result</a>).


structure_id: CompHaus
property_id: disjoint finite coproducts
type: category
proof:
Since it is extensive, it has disjoint finite coproducts (by <a href="/category-implication/extensive_consequences">this result</a>).


structure_id: CompHaus
property_id: coaccessible
type: category
proof:
Since it is locally copresentable, it is coaccessible (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>).


structure_id: CompHaus
property_id: cofiltered limits
type: category
proof:
Since it has cofiltered-limit-stable epimorphisms, it has cofiltered limits (by <a href="/category-implication/dual_filtered_monos_assumption">this result</a>).


structure_id: CompHaus
property_id: cofiltered
type: category
proof:
Since it has equalizers and is semi-strongly connected, it is cofiltered (by <a href="/category-implication/dual_filtered_via_coequalizers">this result</a>).


structure_id: CompHaus
property_id: finitely complete
type: category
proof:
Since it is complete, it is finitely complete (by <a href="/category-implication/complete_consequence">this result</a>).


structure_id: CompHaus
property_id: mono-regular
type: category
proof:
Since it has effective congruences and is extensive, it is mono-regular (by <a href="/category-implication/pretopos_balanced">this result</a>).


structure_id: CompHaus
property_id: initial object
type: category
proof:
Since it has a strict initial object, it has an initial object (by <a href="/category-implication/strict_initial_consequence">this result</a>).


structure_id: CompHaus
property_id: well-copowered
type: category
proof:
Since it is coaccessible, it is well-copowered (by <a href="/category-implication/dual_accessible_well-powered">this result</a>).


structure_id: CompHaus
property_id: finite products
type: category
proof:
Since it is finitely complete, it has finite products (by <a href="/category-implication/finitely_complete_characterization">this result</a>).


structure_id: CompHaus
property_id: filtered colimits
type: category
proof:
Since it has sifted colimits, it has filtered colimits (by <a href="/category-implication/sifted_colimits_consequence">this result</a>).


structure_id: CompHaus
property_id: ℵ₁-cofiltered
type: category
proof:
Since it has countable products and has equalizers, it is ℵ₁-cofiltered (by <a href="/category-implication/dual_aleph1-filtered_criterion">this result</a>).


structure_id: CompHaus
property_id: terminal object
type: category
proof:
Since it is connected and has a multi-terminal object, it has a terminal object (by <a href="/category-implication/multi-terminal_with_connected">this result</a>).


structure_id: CompHaus
property_id: epi-regular
type: category
proof:
Since it is balanced and is regular, it is epi-regular (by <a href="/category-implication/regular_balanced_epi-regular">this result</a>).


structure_id: CompHaus
property_id: distributive
type: category
proof:
Since it is extensive and has finite products, it is distributive (by <a href="/category-implication/extensive_distributivity">this result</a>).


structure_id: CompHaus
property_id: pullbacks
type: category
proof:
Since it has wide pullbacks, it has pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>).


structure_id: CompHaus
property_id: sequential colimits
type: category
proof:
Since it has coequalizers and has countable coproducts, it has sequential colimits (by <a href="/category-implication/dual_sequential_limits_criterion">this result</a>).


structure_id: CompHaus
property_id: cocartesian cofiltered limits
type: category
proof:
Since it has cofiltered limits and is extensive and has a terminal object, it has cocartesian cofiltered limits (by <a href="/category-implication/extensive_cocartesian_cofiltered_limits">this result</a>).


structure_id: CompHaus
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would be coextensive and would be coregular and would be mono-regular, so it would be Malcev (by <a href="/category-implication/dual_regular_epi-regular_extensive_consequences">this result</a>) – contradiction.


structure_id: CompHaus
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be co-Malcev and would be self-dual, so it would be Malcev (by <a href="/category-implication/self-dual_co-Malcev">this result</a>) – contradiction.


structure_id: CompHaus
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have countable coproducts, so it would be countably distributive (by <a href="/category-implication/countably_distributive_criterion">this result</a>) – contradiction.


structure_id: CompHaus
property_id: cartesian filtered colimits
type: category
proof:
Assume for contradiction that it has cartesian filtered colimits. Then it would have cartesian filtered colimits and would have countable coproducts and would be distributive, so it would be countably distributive (by <a href="/category-implication/countably_distributive_filtered_criterion">this result</a>) – contradiction.


structure_id: CompHaus
property_id: locally presentable
type: category
proof:
Assume for contradiction that it is locally presentable. Then it would be locally copresentable and would be locally presentable, so it would be essentially small (by <a href="/category-implication/locally_presentable_essentially_small">this result</a>) – contradiction.


structure_id: CompHaus
property_id: strongly connected
type: category
proof:
Assume for contradiction that it is strongly connected. Then it would have finite products and would be strongly connected, so it would have disjoint finite products (by <a href="/category-implication/strongly_connected_disjoint_products">this result</a>) – contradiction.


structure_id: CompHaus
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have biproducts and would have filtered colimits, so it would have cartesian filtered colimits (by <a href="/category-implication/biproducts_cartesian_filtered_colimits">this result</a>) – contradiction.


structure_id: CompHaus
property_id: pointed
type: category
proof:
Assume for contradiction that it is pointed. Then it would be pointed and would have a strict initial object, so it would be trivial (by <a href="/category-implication/strict_initial_trivial">this result</a>) – contradiction.


structure_id: CompHaus
property_id: disjoint products
type: category
proof:
Assume for contradiction that it has disjoint products. Then it would have disjoint finite products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: CompHaus
property_id: accessible
type: category
proof:
Assume for contradiction that it is accessible. Then it would be accessible and would be cocomplete, so it would be locally presentable (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


structure_id: CompHaus
property_id: locally ℵ₁-presentable
type: category
proof:
Assume for contradiction that it is locally ℵ₁-presentable. Then it would be locally presentable (by <a href="/category-implication/locally_countably_presentable_raise">this result</a>) – contradiction.


structure_id: CompHaus
property_id: ℵ₁-accessible
type: category
proof:
Assume for contradiction that it is ℵ₁-accessible. Then it would be cocomplete and would be ℵ₁-accessible, so it would be locally ℵ₁-presentable (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>) – contradiction.


structure_id: CompHaus
property_id: locally multi-presentable
type: category
proof:
Assume for contradiction that it is locally multi-presentable. Then it would be accessible (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>) – contradiction.


structure_id: CompHaus
property_id: locally poly-presentable
type: category
proof:
Assume for contradiction that it is locally poly-presentable. Then it would be accessible (by <a href="/category-implication/locally_poly-presentable_definition">this result</a>) – contradiction.


structure_id: Grp_c
property_id: unital
type: category
proof:
Since it is Malcev and is pointed, it is unital (by <a href="/category-implication/malcev_implies_unital">this result</a>).


structure_id: Grp_c
property_id: zero morphisms
type: category
proof:
Since it is pointed, it has zero morphisms (by <a href="/category-implication/pointed_characterization">this result</a>).


structure_id: Grp_c
property_id: finite coproducts
type: category
proof:
Since it has countable coproducts, it has finite coproducts (by <a href="/category-implication/dual_countable_products_consequence">this result</a>).


structure_id: Grp_c
property_id: cogenerating set
type: category
proof:
Since it is essentially small, it has a cogenerating set (by <a href="/category-implication/dual_essentially_small_consequences">this result</a>).


structure_id: Grp_c
property_id: accessible
type: category
proof:
Since it is Cauchy complete and is essentially small, it is accessible (by <a href="/category-implication/small_accessible_characterization">this result</a>).


structure_id: Grp_c
property_id: finitely cocomplete
type: category
proof:
Since it has coequalizers and has finite coproducts, it is finitely cocomplete (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>).


structure_id: Grp_c
property_id: abelian
type: category
proof:
Assume for contradiction that it is abelian. Then it would be normal (by <a href="/category-implication/abelian_definition">this result</a>) – contradiction.


structure_id: Grp_c
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have countable copowers, so it would have countable powers (by <a href="/category-implication/countable_power_construction">this result</a>) – contradiction.


structure_id: Grp_c
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be additive and would have effective congruences, so it would be normal (by <a href="/category-implication/additive_effective_congruences_imply_normal">this result</a>) – contradiction.


structure_id: Grp_c
property_id: disjoint coproducts
type: category
proof:
Assume for contradiction that it has disjoint coproducts. Then it would have a cogenerating set and would have disjoint coproducts, so it would have a cogenerator (by <a href="/category-implication/disjoint_coproduct_cogenerator">this result</a>) – contradiction.


structure_id: Grp_c
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would be mono-regular and would be preadditive, so it would be normal (by <a href="/category-implication/normal_criterion">this result</a>) – contradiction.


structure_id: Grp_c
property_id: countable products
type: category
proof:
Assume for contradiction that it has countable products. Then it would have countable powers (by <a href="/category-implication/countable_products_consequence">this result</a>) – contradiction.


structure_id: Grp_c
property_id: subobject classifier
type: category
proof:
Assume for contradiction that it has a subobject classifier. Then it would be pointed and would have a subobject classifier, so it would be normal (by <a href="/category-implication/subobject_classifier_pointed_case">this result</a>) – contradiction.


structure_id: Grp_c
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have biproducts and would be finitely cocomplete, so it would be counital (by <a href="/category-implication/dual_biproducts_unital">this result</a>) – contradiction.


structure_id: Grp_c
property_id: cofiltered limits
type: category
proof:
Assume for contradiction that it has cofiltered limits. Then it would have ℵ₁-cofiltered limits (by <a href="/category-implication/dual_aleph1-filtered-colimits-include-filtered-colimits">this result</a>) – contradiction.


structure_id: Grp_c
property_id: products
type: category
proof:
Assume for contradiction that it has products. Then it would have a cogenerating set and would have products and would have zero morphisms, so it would have a cogenerator (by <a href="/category-implication/dual_generator_via_coproduct">this result</a>) – contradiction.


structure_id: Grp_c
property_id: ℵ₂-small coproducts
type: category
proof:
Assume for contradiction that it has ℵ₂-small coproducts. Then it would have ℵ₂-small copowers (by <a href="/category-implication/dual_aleph2-small_products_consequences">this result</a>) – contradiction.


structure_id: Grp_c
property_id: copowers
type: category
proof:
Assume for contradiction that it has copowers. Then it would have ℵ₂-small copowers (by <a href="/category-implication/dual_powers_include_aleph2-small_powers">this result</a>) – contradiction.


structure_id: Grp_c
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be inhabited and would be thin, so it would have a cogenerator (by <a href="/category-implication/dual_thin_inhabited_consequence">this result</a>) – contradiction.


structure_id: Grp_c
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be conormal and would be self-dual, so it would be normal (by <a href="/category-implication/self-dual_conormal">this result</a>) – contradiction.


structure_id: Grp_c
property_id: extensive
type: category
proof:
Assume for contradiction that it is extensive. Then it would be epi-regular and would be extensive and would be regular, so it would be co-Malcev (by <a href="/category-implication/regular_epi-regular_extensive_consequences">this result</a>) – contradiction.


structure_id: Grp_c
property_id: coproducts
type: category
proof:
Assume for contradiction that it has coproducts. Then it would have coproducts and would have disjoint finite coproducts, so it would have disjoint coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: Grp_c
property_id: CIP
type: category
proof:
Assume for contradiction that it satisfies CIP. Then it would have products (by <a href="/category-implication/CIP_assumption">this result</a>) – contradiction.


structure_id: Grp_c
property_id: directed limits
type: category
proof:
Assume for contradiction that it has directed limits. Then it would have sequential limits (by <a href="/category-implication/directed_limits_consequence">this result</a>) – contradiction.


structure_id: Grp_c
property_id: powers
type: category
proof:
Assume for contradiction that it has powers. Then it would have ℵ₂-small powers (by <a href="/category-implication/powers_include_aleph2-small_powers">this result</a>) – contradiction.


structure_id: Grp_c
property_id: ℵ₁-filtered colimits
type: category
proof:
Assume for contradiction that it has ℵ₁-filtered colimits. Then it would have countable copowers and would have ℵ₁-filtered colimits, so it would have copowers (by <a href="/category-implication/copowers_criterion_countable">this result</a>) – contradiction.


structure_id: Grp_c
property_id: regular subobject classifier
type: category
proof:
Assume for contradiction that it has a regular subobject classifier. Then it would be mono-regular and would have a regular subobject classifier, so it would have a subobject classifier (by <a href="/category-implication/regular_subobjects_suffice">this result</a>) – contradiction.


structure_id: Grp_c
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be right cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/dual_zero-no-mono">this result</a>) – contradiction.


structure_id: Grp_c
property_id: disjoint products
type: category
proof:
Assume for contradiction that it has disjoint products. Then it would have products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: Grp_c
property_id: countably codistributive
type: category
proof:
Assume for contradiction that it is countably codistributive. Then it would have countable products (by <a href="/category-implication/dual_countably_distributive_assumption">this result</a>) – contradiction.


structure_id: Grp_c
property_id: CSP
type: category
proof:
Assume for contradiction that it satisfies CSP. Then it would have products (by <a href="/category-implication/dual_CIP_assumption">this result</a>) – contradiction.


structure_id: Grp_c
property_id: locally finite
type: category
proof:
Assume for contradiction that it is locally finite. Then it would have countable copowers and would be locally finite, so it would be thin (by <a href="/category-implication/dual_thin_via_locally-finite">this result</a>) – contradiction.


structure_id: Grp_c
property_id: essentially countable
type: category
proof:
Assume for contradiction that it is essentially countable. Then it would have countable copowers and would be essentially countable, so it would be thin (by <a href="/category-implication/dual_freyd_countable">this result</a>) – contradiction.


structure_id: Grp_c
property_id: ℵ₁-accessible
type: category
proof:
Assume for contradiction that it is ℵ₁-accessible. Then it would have ℵ₁-filtered colimits (by <a href="/category-implication/countably_accessible_special_case">this result</a>) – contradiction.


structure_id: Grp_c
property_id: locally presentable
type: category
proof:
Assume for contradiction that it is locally presentable. Then it would be complete (by <a href="/category-implication/locally_presentable_another_definition">this result</a>) – contradiction.


structure_id: Grp_c
property_id: infinitary distributive
type: category
proof:
Assume for contradiction that it is infinitary distributive. Then it would have coproducts (by <a href="/category-implication/infinitary_distributive_assumption">this result</a>) – contradiction.


structure_id: Grp_c
property_id: infinitary extensive
type: category
proof:
Assume for contradiction that it is infinitary extensive. Then it would have coproducts (by <a href="/category-implication/infinitary_extensive_assumption">this result</a>) – contradiction.


structure_id: Grp_c
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would have coproducts (by <a href="/category-implication/dual_complete_characterization">this result</a>) – contradiction.


structure_id: Grp_c
property_id: locally finitely multi-presentable
type: category
proof:
Assume for contradiction that it is locally finitely multi-presentable. Then it would have connected limits (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>) – contradiction.


structure_id: Set_c
property_id: natural numbers object
type: category
proof:
Since it is countably distributive, it has a natural numbers object (by <a href="/category-implication/nno_criterion">this result</a>).


structure_id: Set_c
property_id: countable coproducts
type: category
proof:
Since it is countably distributive, it has countable coproducts (by <a href="/category-implication/countably_distributive_assumption">this result</a>).


structure_id: Set_c
property_id: distributive
type: category
proof:
Since it is countably distributive, it is distributive (by <a href="/category-implication/countably_distributive_trivial">this result</a>).


structure_id: Set_c
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have countable copowers and would be self-dual, so it would have countable powers (by <a href="/category-implication/self-dual_countable copowers">this result</a>) – contradiction.


structure_id: Set_c
property_id: trivial
type: category
proof:
Assume for contradiction that it is trivial. Then it would be self-dual (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


structure_id: Set_c
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be Cauchy complete and would be essentially finite, so it would have cofiltered limits (by <a href="/category-implication/dual_finite_filtered_colimits">this result</a>) – contradiction.


structure_id: Set_c
property_id: coproducts
type: category
proof:
Assume for contradiction that it has coproducts. Then it would have copowers (by <a href="/category-implication/dual_products_include_powers">this result</a>) – contradiction.


structure_id: Set_c
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have an initial object and would be thin, so it would have copowers (by <a href="/category-implication/dual_thin_power">this result</a>) – contradiction.


structure_id: Set_c
property_id: pointed
type: category
proof:
Assume for contradiction that it is pointed. Then it would have a natural numbers object and would be pointed, so it would be trivial (by <a href="/category-implication/nno_pointed_case">this result</a>) – contradiction.


structure_id: Set_c
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would have coproducts (by <a href="/category-implication/grothendieck_abelian_definition">this result</a>) – contradiction.


structure_id: Set_c
property_id: CIP
type: category
proof:
Assume for contradiction that it satisfies CIP. Then it would have coproducts (by <a href="/category-implication/CIP_assumption">this result</a>) – contradiction.


structure_id: Set_c
property_id: quasitopos
type: category
proof:
Assume for contradiction that it is a quasitopos. Then it would be locally cartesian closed (by <a href="/category-implication/quasitopos_definition">this result</a>) – contradiction.


structure_id: Set_c
property_id: CSP
type: category
proof:
Assume for contradiction that it satisfies CSP. Then it would have coproducts (by <a href="/category-implication/dual_CIP_assumption">this result</a>) – contradiction.


structure_id: Set_c
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would have connected limits (by <a href="/category-implication/complete_consequence">this result</a>) – contradiction.


structure_id: Set_c
property_id: strongly connected
type: category
proof:
Assume for contradiction that it is strongly connected. Then it would have an initial object and would be strongly connected and would have a terminal object, so it would be pointed (by <a href="/category-implication/strongly_connected_pointed_criterion">this result</a>) – contradiction.


structure_id: Set_c
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would have an initial object and would have zero morphisms, so it would be pointed (by <a href="/category-implication/pointed_characterization">this result</a>) – contradiction.


structure_id: Set_c
property_id: locally copresentable
type: category
proof:
Assume for contradiction that it is locally copresentable. Then it would be cocomplete (by <a href="/category-implication/dual_locally_presentable_another_definition">this result</a>) – contradiction.


structure_id: Fld
property_id: generalized variety
type: category
proof:
Since it is multi-algebraic, it is a generalized variety (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>).


structure_id: Fld
property_id: Cauchy complete
type: category
proof:
Since it is left cancellative, it is Cauchy complete (by <a href="/category-implication/cauchy_complete_criterion">this result</a>).


structure_id: Fld
property_id: coreflexive equalizers
type: category
proof:
Since it is left cancellative, it has coreflexive equalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>).


structure_id: Fld
property_id: effective cocongruences
type: category
proof:
Since it is left cancellative, it has effective cocongruences (by <a href="/category-implication/reflexive_pair_trivial">this result</a>).


structure_id: Fld
property_id: reflexive coequalizers
type: category
proof:
Since it is left cancellative, it has reflexive coequalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>).


structure_id: Fld
property_id: regular-quotient-trivial
type: category
proof:
Since it is left cancellative, it is regular-quotient-trivial (by <a href="/category-implication/dual_regular-subobject-trivial_criterion">this result</a>).


structure_id: Fld
property_id: finitely accessible
type: category
proof:
Since it is locally finitely multi-presentable, it is finitely accessible (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>).


structure_id: Fld
property_id: equalizers
type: category
proof:
Since it has connected limits, it has equalizers (by <a href="/category-implication/connected_limits_characterization">this result</a>).


structure_id: Fld
property_id: multi-complete
type: category
proof:
Assume for contradiction that it is multi-complete. Then it would have a multi-terminal object (by <a href="/category-implication/multi-terminal_special_case">this result</a>) – contradiction.


structure_id: Fld
property_id: terminal object
type: category
proof:
Assume for contradiction that it has a terminal object. Then it would be connected (by <a href="/category-implication/multi-terminal_with_connected">this result</a>) – contradiction.


structure_id: Fld
property_id: semi-strongly connected
type: category
proof:
Assume for contradiction that it is semi-strongly connected. Then it would be connected (by <a href="/category-implication/semi-strongly_connected_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: sifted
type: category
proof:
Assume for contradiction that it is sifted. Then it would be connected (by <a href="/category-implication/sifted_is_connected">this result</a>) – contradiction.


structure_id: Fld
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be locally cartesian closed (by <a href="/category-implication/groupoid_lcc">this result</a>) – contradiction.


structure_id: Fld
property_id: binary products
type: category
proof:
Assume for contradiction that it has binary products. Then it would have binary powers (by <a href="/category-implication/binary_products_include_binary_powers">this result</a>) – contradiction.


structure_id: Fld
property_id: finite powers
type: category
proof:
Assume for contradiction that it has finite powers. Then it would have binary powers (by <a href="/category-implication/finite_powers_consequences">this result</a>) – contradiction.


structure_id: Fld
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be locally finite (by <a href="/category-implication/essentially_finite_raise">this result</a>) – contradiction.


structure_id: Fld
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be locally finite (by <a href="/category-implication/thin_consequences">this result</a>) – contradiction.


structure_id: Fld
property_id: gaunt
type: category
proof:
Assume for contradiction that it is gaunt. Then it would be core-thin (by <a href="/category-implication/gaunt_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would be locally cartesian closed (by <a href="/category-implication/topos_is_locally_cartesian_closed">this result</a>) – contradiction.


structure_id: Fld
property_id: locally cocartesian coclosed
type: category
proof:
Assume for contradiction that it is locally cocartesian coclosed. Then it would have pushouts (by <a href="/category-implication/dual_pullbacks_are_local_products">this result</a>) – contradiction.


structure_id: Fld
property_id: initial object
type: category
proof:
Assume for contradiction that it has an initial object. Then it would be connected (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>) – contradiction.


structure_id: Fld
property_id: cosifted
type: category
proof:
Assume for contradiction that it is cosifted. Then it would be connected (by <a href="/category-implication/dual_sifted_is_connected">this result</a>) – contradiction.


structure_id: Fld
property_id: wide pushouts
type: category
proof:
Assume for contradiction that it has wide pushouts. Then it would have pushouts (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>) – contradiction.


structure_id: Fld
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have filtered-colimit-stable monomorphisms and would be self-dual, so it would have cofiltered-limit-stable epimorphisms (by <a href="/category-implication/self-dual_filtered-colimit-stable monomorphisms">this result</a>) – contradiction.


structure_id: Fld
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would be left cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/zero-no-mono">this result</a>) – contradiction.


structure_id: Fld
property_id: binary copowers
type: category
proof:
Assume for contradiction that it has binary copowers. Then it would have binary copowers and would be left cancellative, so it would be thin (by <a href="/category-implication/codiagonal-no-mono">this result</a>) – contradiction.


structure_id: Fld
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would be multi-complete (by <a href="/category-implication/multi-complete_generalize_limits">this result</a>) – contradiction.


structure_id: Fld
property_id: filtered
type: category
proof:
Assume for contradiction that it is filtered. Then it would be sifted (by <a href="/category-implication/filtered_is_sifted">this result</a>) – contradiction.


structure_id: Fld
property_id: binary coproducts
type: category
proof:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would be inhabited, so it would be sifted (by <a href="/category-implication/coproducts_implies_sifted">this result</a>) – contradiction.


structure_id: Fld
property_id: pointed
type: category
proof:
Assume for contradiction that it is pointed. Then it would have an initial object (by <a href="/category-implication/pointed_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: strict initial object
type: category
proof:
Assume for contradiction that it has a strict initial object. Then it would have an initial object (by <a href="/category-implication/strict_initial_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: finite products
type: category
proof:
Assume for contradiction that it has finite products. Then it would have binary products (by <a href="/category-implication/finite_products_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: countable powers
type: category
proof:
Assume for contradiction that it has countable powers. Then it would have finite powers (by <a href="/category-implication/countable_powers_include_finite_powers">this result</a>) – contradiction.


structure_id: Fld
property_id: locally copresentable
type: category
proof:
Assume for contradiction that it is locally copresentable. Then it would be coaccessible (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>) – contradiction.


structure_id: Fld
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would be coextensive and would have effective cocongruences, so it would be epi-regular (by <a href="/category-implication/dual_pretopos_balanced">this result</a>) – contradiction.


structure_id: Fld
property_id: cofiltered
type: category
proof:
Assume for contradiction that it is cofiltered. Then it would be cosifted (by <a href="/category-implication/dual_filtered_is_sifted">this result</a>) – contradiction.


structure_id: Fld
property_id: strict terminal object
type: category
proof:
Assume for contradiction that it has a strict terminal object. Then it would have a terminal object (by <a href="/category-implication/dual_strict_initial_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: finite coproducts
type: category
proof:
Assume for contradiction that it has finite coproducts. Then it would have an initial object (by <a href="/category-implication/dual_finite_products_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: finite copowers
type: category
proof:
Assume for contradiction that it has finite copowers. Then it would have an initial object (by <a href="/category-implication/dual_finite_powers_consequences">this result</a>) – contradiction.


structure_id: Fld
property_id: connected colimits
type: category
proof:
Assume for contradiction that it has connected colimits. Then it would have wide pushouts (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: quotient object classifier
type: category
proof:
Assume for contradiction that it has a quotient object classifier. Then it would be epi-regular (by <a href="/category-implication/dual_subobject_classifier_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: coequalizers
type: category
proof:
Assume for contradiction that it has coequalizers. Then it would have coequalizers and would be regular-quotient-trivial, so it would be thin (by <a href="/category-implication/dual_thin_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: Fld
property_id: natural numbers object
type: category
proof:
Assume for contradiction that it has a natural numbers object. Then it would have finite products (by <a href="/category-implication/nno_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would have finite products (by <a href="/category-implication/additive_definition">this result</a>) – contradiction.


structure_id: Fld
property_id: abelian
type: category
proof:
Assume for contradiction that it is abelian. Then it would be conormal (by <a href="/category-implication/abelian_definition">this result</a>) – contradiction.


structure_id: Fld
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have finite coproducts (by <a href="/category-implication/biproducts_definition_consequences">this result</a>) – contradiction.


structure_id: Fld
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would have finite products (by <a href="/category-implication/ccc_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: products
type: category
proof:
Assume for contradiction that it has products. Then it would have equalizers and would have products, so it would be complete (by <a href="/category-implication/complete_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: finitely complete
type: category
proof:
Assume for contradiction that it is finitely complete. Then it would have finite products (by <a href="/category-implication/finitely_complete_characterization">this result</a>) – contradiction.


structure_id: Fld
property_id: disjoint finite coproducts
type: category
proof:
Assume for contradiction that it has disjoint finite coproducts. Then it would have finite coproducts (by <a href="/category-implication/disjoint_coproducts_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: infinitary distributive
type: category
proof:
Assume for contradiction that it is infinitary distributive. Then it would have finite products (by <a href="/category-implication/infinitary_distributive_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: countably distributive
type: category
proof:
Assume for contradiction that it is countably distributive. Then it would have finite products (by <a href="/category-implication/countably_distributive_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: distributive
type: category
proof:
Assume for contradiction that it is distributive. Then it would have finite coproducts (by <a href="/category-implication/distributive_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: cartesian filtered colimits
type: category
proof:
Assume for contradiction that it has cartesian filtered colimits. Then it would have finite products (by <a href="/category-implication/cartesian_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: finitely cocomplete
type: category
proof:
Assume for contradiction that it is finitely cocomplete. Then it would be filtered (by <a href="/category-implication/filtered_criterion">this result</a>) – contradiction.


structure_id: Fld
property_id: ℵ₁-filtered
type: category
proof:
Assume for contradiction that it is ℵ₁-filtered. Then it would be filtered (by <a href="/category-implication/aleph1-filtered-consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: cocartesian coclosed
type: category
proof:
Assume for contradiction that it is cocartesian coclosed. Then it would have finite coproducts (by <a href="/category-implication/dual_ccc_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would have connected colimits (by <a href="/category-implication/dual_complete_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: disjoint finite products
type: category
proof:
Assume for contradiction that it has disjoint finite products. Then it would have finite products (by <a href="/category-implication/dual_disjoint_coproducts_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: infinitary codistributive
type: category
proof:
Assume for contradiction that it is infinitary codistributive. Then it would have finite coproducts (by <a href="/category-implication/dual_infinitary_distributive_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: countably codistributive
type: category
proof:
Assume for contradiction that it is countably codistributive. Then it would have finite coproducts (by <a href="/category-implication/dual_countably_distributive_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: codistributive
type: category
proof:
Assume for contradiction that it is codistributive. Then it would have finite coproducts (by <a href="/category-implication/dual_distributive_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: cocartesian cofiltered limits
type: category
proof:
Assume for contradiction that it has cocartesian cofiltered limits. Then it would have finite coproducts (by <a href="/category-implication/dual_cartesian_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: ℵ₁-cofiltered
type: category
proof:
Assume for contradiction that it is ℵ₁-cofiltered. Then it would be cofiltered (by <a href="/category-implication/dual_aleph1-filtered-consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: countable coproducts
type: category
proof:
Assume for contradiction that it has countable coproducts. Then it would have finite coproducts (by <a href="/category-implication/dual_countable_products_consequence">this result</a>) – contradiction.


structure_id: Fld
property_id: countable copowers
type: category
proof:
Assume for contradiction that it has countable copowers. Then it would have finite copowers (by <a href="/category-implication/dual_countable_powers_include_finite_powers">this result</a>) – contradiction.


structure_id: Fld
property_id: Malcev
type: category
proof:
Assume for contradiction that it is Malcev. Then it would be finitely complete (by <a href="/category-implication/malcev_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: regular
type: category
proof:
Assume for contradiction that it is regular. Then it would be finitely complete (by <a href="/category-implication/regular_def">this result</a>) – contradiction.


structure_id: Fld
property_id: exact filtered colimits
type: category
proof:
Assume for contradiction that it has exact filtered colimits. Then it would be finitely complete (by <a href="/category-implication/exact_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: regular subobject classifier
type: category
proof:
Assume for contradiction that it has a regular subobject classifier. Then it would be finitely complete (by <a href="/category-implication/regular_subobject_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: co-Malcev
type: category
proof:
Assume for contradiction that it is co-Malcev. Then it would be finitely cocomplete (by <a href="/category-implication/dual_malcev_condition">this result</a>) – contradiction.


structure_id: Fld
property_id: coregular
type: category
proof:
Assume for contradiction that it is coregular. Then it would be finitely cocomplete (by <a href="/category-implication/dual_regular_def">this result</a>) – contradiction.


structure_id: Fld
property_id: regular quotient object classifier
type: category
proof:
Assume for contradiction that it has a regular quotient object classifier. Then it would be finitely cocomplete (by <a href="/category-implication/dual_regular_subobject_assumption">this result</a>) – contradiction.


structure_id: Fld
property_id: Barr-exact
type: category
proof:
Assume for contradiction that it is Barr-exact. Then it would be regular (by <a href="/category-implication/Barr-exact_definition">this result</a>) – contradiction.


structure_id: Fld
property_id: coproducts
type: category
proof:
Assume for contradiction that it has coproducts. Then it would have ℵ₂-small coproducts (by <a href="/category-implication/dual_products_include_aleph2-small_products">this result</a>) – contradiction.


structure_id: FiltVect
property_id: ℵ₁-accessible
type: category
proof:
Since it is finitely accessible, it is ℵ₁-accessible (by <a href="/category-implication/finitely_accessible_raise">this result</a>).


structure_id: FiltVect
property_id: locally essentially small
type: category
proof:
Since it is preadditive, it is locally essentially small (by <a href="/category-implication/preadditive_consequence">this result</a>).


structure_id: FiltVect
property_id: zero morphisms
type: category
proof:
Since it is preadditive, it has zero morphisms (by <a href="/category-implication/preadditive_consequence">this result</a>).


structure_id: FiltVect
property_id: equalizers
type: category
proof:
Since it has kernels and is preadditive, it has equalizers (by <a href="/category-implication/equalizers_via_kernels">this result</a>).


structure_id: FiltVect
property_id: coequalizers
type: category
proof:
Since it has cokernels and is preadditive, it has coequalizers (by <a href="/category-implication/dual_equalizers_via_kernels">this result</a>).


structure_id: FiltVect
property_id: cocomplete
type: category
proof:
Since it has coequalizers and has coproducts, it is cocomplete (by <a href="/category-implication/dual_complete_characterization">this result</a>).


structure_id: FiltVect
property_id: locally finitely presentable
type: category
proof:
Since it is cocomplete and is finitely accessible, it is locally finitely presentable (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>).


structure_id: FiltVect
property_id: locally ℵ₁-presentable
type: category
proof:
Since it is cocomplete and is ℵ₁-accessible, it is locally ℵ₁-presentable (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>).


structure_id: FiltVect
property_id: locally presentable
type: category
proof:
Since it is accessible and is cocomplete, it is locally presentable (by <a href="/category-implication/locally_presentable_definition">this result</a>).


structure_id: FiltVect
property_id: additive
type: category
proof:
Since it has finite products and is preadditive, it is additive (by <a href="/category-implication/additive_definition">this result</a>).


structure_id: FiltVect
property_id: locally finitely multi-presentable
type: category
proof:
Since it has connected limits and is finitely accessible, it is locally finitely multi-presentable (by <a href="/category-implication/locally_finitely_multi-presentable_definition">this result</a>).


structure_id: FiltVect
property_id: pointed
type: category
proof:
Since it has an initial object and is strongly connected and has a terminal object, it is pointed (by <a href="/category-implication/strongly_connected_pointed_criterion">this result</a>).


structure_id: FiltVect
property_id: counital
type: category
proof:
Since it is co-Malcev and is pointed, it is counital (by <a href="/category-implication/dual_malcev_implies_unital">this result</a>).


structure_id: FiltVect
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be mono-regular (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


structure_id: FiltVect
property_id: effective congruences
type: category
proof:
Assume for contradiction that it has effective congruences. Then it would be additive and would have effective congruences, so it would be normal (by <a href="/category-implication/additive_effective_congruences_imply_normal">this result</a>) – contradiction.


structure_id: FiltVect
property_id: effective cocongruences
type: category
proof:
Assume for contradiction that it has effective cocongruences. Then it would be additive and would have effective cocongruences, so it would be conormal (by <a href="/category-implication/dual_additive_effective_congruences_imply_normal">this result</a>) – contradiction.


structure_id: FiltVect
property_id: multi-algebraic
type: category
proof:
Assume for contradiction that it is multi-algebraic. Then it would have effective congruences (by <a href="/category-implication/multi-algebraic_implies_effective_congruences">this result</a>) – contradiction.


structure_id: FiltVect
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would have effective congruences (by <a href="/category-implication/core-hin_quotients">this result</a>) – contradiction.


structure_id: FiltVect
property_id: Barr-exact
type: category
proof:
Assume for contradiction that it is Barr-exact. Then it would have effective congruences (by <a href="/category-implication/Barr-exact_definition">this result</a>) – contradiction.


structure_id: FiltVect
property_id: Barr-coexact
type: category
proof:
Assume for contradiction that it is Barr-coexact. Then it would have effective cocongruences (by <a href="/category-implication/dual_Barr-exact_definition">this result</a>) – contradiction.


structure_id: FiltVect
property_id: locally strongly finitely presentable
type: category
proof:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be multi-algebraic (by <a href="/category-implication/varieties_are_multi-algebraic">this result</a>) – contradiction.


structure_id: FiltVect
property_id: pretopos
type: category
proof:
Assume for contradiction that it is a pretopos. Then it would be Barr-exact (by <a href="/category-implication/pretopos_definition">this result</a>) – contradiction.


structure_id: FinAb
property_id: essentially small
type: category
proof:
Since it is essentially countable, it is essentially small (by <a href="/category-implication/essentially_countable_consequence">this result</a>).


structure_id: FinAb
property_id: ℵ₁-accessible
type: category
proof:
Since it is Cauchy complete and is essentially countable and is locally finite, it is ℵ₁-accessible (by <a href="/category-implication/countably_accessible_criterion">this result</a>).


structure_id: FinAb
property_id: inhabited
type: category
proof:
Since it is connected, it is inhabited (by <a href="/category-implication/connected_consequence">this result</a>).


structure_id: FinAb
property_id: ℵ₁-cofiltered limits
type: category
proof:
Since it is self-dual and has ℵ₁-filtered colimits, it has ℵ₁-cofiltered limits (by <a href="/category-implication/self-dual_ℵ₁-filtered colimits">this result</a>).


structure_id: FinAb
property_id: coproducts
type: category
proof:
Assume for contradiction that it has coproducts. Then it would have coproducts and would have a generating set and would have zero morphisms, so it would have a generator (by <a href="/category-implication/generator_via_coproduct">this result</a>) – contradiction.


structure_id: FinAb
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be inhabited and would be thin, so it would have a generator (by <a href="/category-implication/thin_inhabited_consequence">this result</a>) – contradiction.


structure_id: FinAb
property_id: cogenerator
type: category
proof:
Assume for contradiction that it has a cogenerator. Then it would have a cogenerator and would be self-dual, so it would have a generator (by <a href="/category-implication/self-dual_cogenerator">this result</a>) – contradiction.


structure_id: FinAb
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be a groupoid and would have a multi-terminal object, so it would be thin (by <a href="/category-implication/groupoid_with_multi-terminal">this result</a>) – contradiction.


structure_id: FinAb
property_id: countable powers
type: category
proof:
Assume for contradiction that it has countable powers. Then it would have countable powers and would be locally finite, so it would be thin (by <a href="/category-implication/thin_via_locally-finite">this result</a>) – contradiction.


structure_id: FinAb
property_id: products
type: category
proof:
Assume for contradiction that it has products. Then it would have disjoint finite products and would have products, so it would have disjoint products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: FinAb
property_id: filtered colimits
type: category
proof:
Assume for contradiction that it has filtered colimits. Then it would have filtered colimits and would have finite coproducts, so it would have coproducts (by <a href="/category-implication/dual_products_criterion">this result</a>) – contradiction.


structure_id: FinAb
property_id: countable coproducts
type: category
proof:
Assume for contradiction that it has countable coproducts. Then it would have countable coproducts and would have ℵ₁-filtered colimits, so it would have coproducts (by <a href="/category-implication/dual_products_criterion_countable">this result</a>) – contradiction.


structure_id: FinAb
property_id: countable copowers
type: category
proof:
Assume for contradiction that it has countable copowers. Then it would have countable copowers and would be locally finite, so it would be thin (by <a href="/category-implication/dual_thin_via_locally-finite">this result</a>) – contradiction.


structure_id: FinAb
property_id: cofiltered limits
type: category
proof:
Assume for contradiction that it has cofiltered limits. Then it would have cofiltered limits and would have finite products, so it would have products (by <a href="/category-implication/products_criterion">this result</a>) – contradiction.


structure_id: FinAb
property_id: wide pushouts
type: category
proof:
Assume for contradiction that it has wide pushouts. Then it would have an initial object and would have wide pushouts, so it would be cocomplete (by <a href="/category-implication/dual_complete_criterion">this result</a>) – contradiction.


structure_id: FinAb
property_id: locally multi-presentable
type: category
proof:
Assume for contradiction that it is locally multi-presentable. Then it would be multi-cocomplete (by <a href="/category-implication/locally_multi-presentable_another_definition">this result</a>) – contradiction.


structure_id: FinAb
property_id: multi-algebraic
type: category
proof:
Assume for contradiction that it is multi-algebraic. Then it would be multi-cocomplete (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>) – contradiction.


structure_id: FinAb
property_id: wide pullbacks
type: category
proof:
Assume for contradiction that it has wide pullbacks. Then it would have a terminal object and would have wide pullbacks, so it would be complete (by <a href="/category-implication/complete_criterion">this result</a>) – contradiction.


structure_id: FinAb
property_id: connected limits
type: category
proof:
Assume for contradiction that it has connected limits. Then it would be accessible and would have connected limits, so it would be locally multi-presentable (by <a href="/category-implication/locally_multi-presentable_definition">this result</a>) – contradiction.


structure_id: FinGrp
property_id: directed colimits
type: category
proof:
Assume for contradiction that it has directed colimits. Then it would have sequential colimits (by <a href="/category-implication/dual_directed_limits_consequence">this result</a>) – contradiction.


structure_id: FinGrp
property_id: countable coproducts
type: category
proof:
Assume for contradiction that it has countable coproducts. Then it would have coequalizers and would have countable coproducts, so it would have sequential colimits (by <a href="/category-implication/dual_sequential_limits_criterion">this result</a>) – contradiction.


structure_id: FinGrp
property_id: inverse
type: category
proof:
Assume for contradiction that it is inverse. Then it would have sequential colimits (by <a href="/category-implication/dual_direct_implies_sequential_limits">this result</a>) – contradiction.


structure_id: FinGrp
property_id: binary coproducts
type: category
proof:
Assume for contradiction that it has binary coproducts. Then it would have binary copowers (by <a href="/category-implication/dual_binary_products_include_binary_powers">this result</a>) – contradiction.


structure_id: FinGrp
property_id: finite copowers
type: category
proof:
Assume for contradiction that it has finite copowers. Then it would have binary copowers (by <a href="/category-implication/dual_finite_powers_consequences">this result</a>) – contradiction.


structure_id: FinGrp
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have binary powers and would be self-dual, so it would have binary copowers (by <a href="/category-implication/self-dual_binary powers">this result</a>) – contradiction.


structure_id: FinGrp
property_id: countably distributive
type: category
proof:
Assume for contradiction that it is countably distributive. Then it would have countable coproducts (by <a href="/category-implication/countably_distributive_assumption">this result</a>) – contradiction.


structure_id: FinGrp
property_id: filtered colimits
type: category
proof:
Assume for contradiction that it has filtered colimits. Then it would have directed colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>) – contradiction.


structure_id: FinGrp
property_id: finite coproducts
type: category
proof:
Assume for contradiction that it has finite coproducts. Then it would have binary coproducts (by <a href="/category-implication/dual_finite_products_characterization">this result</a>) – contradiction.


structure_id: FinGrp
property_id: pushouts
type: category
proof:
Assume for contradiction that it has pushouts. Then it would have an initial object and would have pushouts, so it would have binary coproducts (by <a href="/category-implication/dual_binary_products_criterion">this result</a>) – contradiction.


structure_id: FinGrp
property_id: extensive
type: category
proof:
Assume for contradiction that it is extensive. Then it would have finite coproducts (by <a href="/category-implication/extensive_assumption">this result</a>) – contradiction.


structure_id: FinGrp
property_id: sequential limits
type: category
proof:
Assume for contradiction that it has sequential limits. Then it would have finite products and would have sequential limits, so it would have countable products (by <a href="/category-implication/countable_products_criterion">this result</a>) – contradiction.


structure_id: FinGrp
property_id: finitely cocomplete
type: category
proof:
Assume for contradiction that it is finitely cocomplete. Then it would have finite coproducts (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>) – contradiction.


structure_id: FinGrp
property_id: counital
type: category
proof:
Assume for contradiction that it is counital. Then it would be finitely cocomplete (by <a href="/category-implication/dual_unital_assumptions">this result</a>) – contradiction.


structure_id: FinGrp
property_id: quotient object classifier
type: category
proof:
Assume for contradiction that it has a quotient object classifier. Then it would be finitely cocomplete (by <a href="/category-implication/dual_subobject_classifier_consequence">this result</a>) – contradiction.


structure_id: FinOrd
property_id: effective congruences
type: category
proof:
Since it is core-thin, it has effective congruences (by <a href="/category-implication/core-hin_quotients">this result</a>).


structure_id: FinOrd
property_id: quotients of congruences
type: category
proof:
Since it is core-thin, it has quotients of congruences (by <a href="/category-implication/core-hin_quotients">this result</a>).


structure_id: FinOrd
property_id: filtered
type: category
proof:
Since it has coequalizers and is semi-strongly connected, it is filtered (by <a href="/category-implication/filtered_via_coequalizers">this result</a>).


structure_id: FinOrd
property_id: coquotients of cocongruences
type: category
proof:
Since it is core-thin, it has coquotients of cocongruences (by <a href="/category-implication/dual_core-hin_quotients">this result</a>).


structure_id: FinOrd
property_id: effective cocongruences
type: category
proof:
Since it is core-thin, it has effective cocongruences (by <a href="/category-implication/dual_core-hin_quotients">this result</a>).


structure_id: FinOrd
property_id: countable products
type: category
proof:
Assume for contradiction that it has countable products. Then it would have countable products and would have equalizers, so it would have sequential limits (by <a href="/category-implication/sequential_limits_criterion">this result</a>) – contradiction.


structure_id: FinOrd
property_id: direct
type: category
proof:
Assume for contradiction that it is direct. Then it would have sequential limits (by <a href="/category-implication/direct_implies_sequential_limits">this result</a>) – contradiction.


structure_id: FinOrd
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be one-way (by <a href="/category-implication/thin_consequences">this result</a>) – contradiction.


structure_id: FinOrd
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be right cancellative and would have a terminal object, so it would have a strict terminal object (by <a href="/category-implication/dual_strict_initial_left_criterion">this result</a>) – contradiction.


structure_id: FinOrd
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be self-dual and would have a strict initial object, so it would have a strict terminal object (by <a href="/category-implication/self-dual_strict initial object">this result</a>) – contradiction.


structure_id: FinOrd
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be left cancellative and would be sifted, so it would be thin (by <a href="/category-implication/sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


structure_id: FinOrd
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would have directed limits (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


structure_id: FinOrd
property_id: binary powers
type: category
proof:
Assume for contradiction that it has binary powers. Then it would have binary powers and would be core-thin, so it would be thin (by <a href="/category-implication/core-thin_products_thin">this result</a>) – contradiction.


structure_id: FinOrd
property_id: cofiltered limits
type: category
proof:
Assume for contradiction that it has cofiltered limits. Then it would have directed limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>) – contradiction.


structure_id: FinOrd
property_id: binary copowers
type: category
proof:
Assume for contradiction that it has binary copowers. Then it would have binary copowers and would be core-thin, so it would be thin (by <a href="/category-implication/dual_core-thin_products_thin">this result</a>) – contradiction.


structure_id: FinOrd
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be Cauchy complete and would be essentially finite, so it would have filtered colimits (by <a href="/category-implication/finite_filtered_colimits">this result</a>) – contradiction.


structure_id: FinOrd
property_id: locally finitely presentable
type: category
proof:
Assume for contradiction that it is locally finitely presentable. Then it would be finitely accessible (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


structure_id: FinOrd
property_id: pullbacks
type: category
proof:
Assume for contradiction that it has pullbacks. Then it would have pullbacks and would have a terminal object, so it would have binary products (by <a href="/category-implication/binary_products_criterion">this result</a>) – contradiction.


structure_id: FinOrd
property_id: locally cartesian closed
type: category
proof:
Assume for contradiction that it is locally cartesian closed. Then it would have pullbacks (by <a href="/category-implication/pullbacks_are_local_products">this result</a>) – contradiction.


structure_id: FinOrd
property_id: subobject classifier
type: category
proof:
Assume for contradiction that it has a subobject classifier. Then it would be finitely complete (by <a href="/category-implication/subobject_classifier_consequence">this result</a>) – contradiction.


structure_id: FinOrd
property_id: quasitopos
type: category
proof:
Assume for contradiction that it is a quasitopos. Then it would be finitely cocomplete (by <a href="/category-implication/quasitopos_definition">this result</a>) – contradiction.


structure_id: FinSet
property_id: trivial
type: category
proof:
Assume for contradiction that it is trivial. Then it would be essentially discrete (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


structure_id: FinSet
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be cartesian closed and would be self-dual, so it would be cocartesian coclosed (by <a href="/category-implication/self-dual_cartesian closed">this result</a>) – contradiction.


structure_id: B
property_id: directed limits
type: category
proof:
Since it is a groupoid, it has directed limits (by <a href="/category-implication/groupoid_consequence">this result</a>).


structure_id: B
property_id: left cancellative
type: category
proof:
Since it is a groupoid, it is left cancellative (by <a href="/category-implication/groupoid_consequence">this result</a>).


structure_id: B
property_id: mono-regular
type: category
proof:
Since it is a groupoid, it is mono-regular (by <a href="/category-implication/groupoid_consequence">this result</a>).


structure_id: B
property_id: pullbacks
type: category
proof:
Since it is a groupoid, it has pullbacks (by <a href="/category-implication/groupoid_consequence">this result</a>).


structure_id: B
property_id: self-dual
type: category
proof:
Since it is a groupoid, it is self-dual (by <a href="/category-implication/groupoid_consequence">this result</a>).


structure_id: B
property_id: well-powered
type: category
proof:
Since it is a groupoid, it is well-powered (by <a href="/category-implication/groupoid_consequence">this result</a>).


structure_id: B
property_id: locally cartesian closed
type: category
proof:
Since it is a groupoid, it is locally cartesian closed (by <a href="/category-implication/groupoid_lcc">this result</a>).


structure_id: B
property_id: subobject-trivial
type: category
proof:
Since it is a groupoid, it is subobject-trivial (by <a href="/category-implication/subobject-trivial_groupoids">this result</a>).


structure_id: B
property_id: directed colimits
type: category
proof:
Since it is a groupoid, it has directed colimits (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


structure_id: B
property_id: epi-regular
type: category
proof:
Since it is a groupoid, it is epi-regular (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


structure_id: B
property_id: pushouts
type: category
proof:
Since it is a groupoid, it has pushouts (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


structure_id: B
property_id: right cancellative
type: category
proof:
Since it is a groupoid, it is right cancellative (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


structure_id: B
property_id: well-copowered
type: category
proof:
Since it is a groupoid, it is well-copowered (by <a href="/category-implication/dual_groupoid_consequence">this result</a>).


structure_id: B
property_id: locally cocartesian coclosed
type: category
proof:
Since it is a groupoid, it is locally cocartesian coclosed (by <a href="/category-implication/dual_groupoid_lcc">this result</a>).


structure_id: B
property_id: quotient-trivial
type: category
proof:
Since it is a groupoid, it is quotient-trivial (by <a href="/category-implication/dual_subobject-trivial_groupoids">this result</a>).


structure_id: B
property_id: finitely accessible
type: category
proof:
Since it is essentially small and is a groupoid, it is finitely accessible (by <a href="/category-implication/groupoids_are_generalized_varieties">this result</a>).


structure_id: B
property_id: generalized variety
type: category
proof:
Since it is essentially small and is a groupoid, it is a generalized variety (by <a href="/category-implication/groupoids_are_generalized_varieties">this result</a>).


structure_id: B
property_id: effective congruences
type: category
proof:
Since it is left cancellative, it has effective congruences (by <a href="/category-implication/reflexive_pair_trivial">this result</a>).


structure_id: B
property_id: filtered colimits
type: category
proof:
Since it has directed colimits, it has filtered colimits (by <a href="/category-implication/directed_colimits_suffice">this result</a>).


structure_id: B
property_id: regular-subobject-trivial
type: category
proof:
Since it is subobject-trivial, it is regular-subobject-trivial (by <a href="/category-implication/subobject-trivial_characterization">this result</a>).


structure_id: B
property_id: cofiltered limits
type: category
proof:
Since it has directed limits, it has cofiltered limits (by <a href="/category-implication/dual_directed_colimits_suffice">this result</a>).


structure_id: B
property_id: regular-quotient-trivial
type: category
proof:
Since it is quotient-trivial, it is regular-quotient-trivial (by <a href="/category-implication/dual_subobject-trivial_characterization">this result</a>).


structure_id: B
property_id: filtered-colimit-stable monomorphisms
type: category
proof:
Since it has filtered colimits and is left cancellative, it has filtered-colimit-stable monomorphisms (by <a href="/category-implication/filtered_monos_trivial">this result</a>).


structure_id: B
property_id: wide pullbacks
type: category
proof:
Since it has cofiltered limits and has pullbacks, it has wide pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>).


structure_id: B
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Since it has cofiltered limits and is right cancellative, it has cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_filtered_monos_trivial">this result</a>).


structure_id: B
property_id: cosifted limits
type: category
proof:
Since it has cofiltered limits and has coreflexive equalizers and has pushouts, it has cosifted limits (by <a href="/category-implication/dual_sifted_colimits_criterion">this result</a>).


structure_id: B
property_id: wide pushouts
type: category
proof:
Since it has filtered colimits and has pushouts, it has wide pushouts (by <a href="/category-implication/dual_wide_pullbacks_criterion">this result</a>).


structure_id: B
property_id: multi-terminal object
type: category
proof:
Assume for contradiction that it has a multi-terminal object. Then it would be a groupoid and would have a multi-terminal object, so it would be thin (by <a href="/category-implication/groupoid_with_multi-terminal">this result</a>) – contradiction.


structure_id: B
property_id: finite products
type: category
proof:
Assume for contradiction that it has finite products. Then it would have a terminal object (by <a href="/category-implication/finite_products_characterization">this result</a>) – contradiction.


structure_id: B
property_id: finite powers
type: category
proof:
Assume for contradiction that it has finite powers. Then it would have a terminal object (by <a href="/category-implication/finite_powers_consequences">this result</a>) – contradiction.


structure_id: B
property_id: equalizers
type: category
proof:
Assume for contradiction that it has equalizers. Then it would have equalizers and would be regular-subobject-trivial, so it would be thin (by <a href="/category-implication/thin_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: B
property_id: disjoint finite coproducts
type: category
proof:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would be regular-subobject-trivial, so it would be trivial (by <a href="/category-implication/disjoint_finite_coproducts_disallows_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: B
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would be core-thin and would be a groupoid, so it would be thin (by <a href="/category-implication/thin_groupoids">this result</a>) – contradiction.


structure_id: B
property_id: binary powers
type: category
proof:
Assume for contradiction that it has binary powers. Then it would have binary powers and would be right cancellative, so it would be thin (by <a href="/category-implication/dual_codiagonal-no-mono">this result</a>) – contradiction.


structure_id: B
property_id: binary products
type: category
proof:
Assume for contradiction that it has binary products. Then it would have binary products and would be inhabited, so it would be cosifted (by <a href="/category-implication/dual_coproducts_implies_sifted">this result</a>) – contradiction.


structure_id: B
property_id: multi-initial object
type: category
proof:
Assume for contradiction that it has a multi-initial object. Then it would be a groupoid and would have a multi-initial object, so it would be thin (by <a href="/category-implication/dual_groupoid_with_multi-terminal">this result</a>) – contradiction.


structure_id: B
property_id: disjoint finite products
type: category
proof:
Assume for contradiction that it has disjoint finite products. Then it would have disjoint finite products and would be regular-quotient-trivial, so it would be trivial (by <a href="/category-implication/dual_disjoint_finite_coproducts_disallows_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: B
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would have equalizers (by <a href="/category-implication/complete_characterization">this result</a>) – contradiction.


structure_id: B
property_id: finitely complete
type: category
proof:
Assume for contradiction that it is finitely complete. Then it would have equalizers (by <a href="/category-implication/finitely_complete_characterization">this result</a>) – contradiction.


structure_id: B
property_id: connected limits
type: category
proof:
Assume for contradiction that it has connected limits. Then it would have equalizers (by <a href="/category-implication/connected_limits_characterization">this result</a>) – contradiction.


structure_id: B
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would have disjoint finite coproducts (by <a href="/category-implication/topos_consequence">this result</a>) – contradiction.


structure_id: B
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would have coequalizers (by <a href="/category-implication/dual_complete_characterization">this result</a>) – contradiction.


structure_id: B
property_id: multi-cocomplete
type: category
proof:
Assume for contradiction that it is multi-cocomplete. Then it would have a multi-initial object (by <a href="/category-implication/dual_multi-terminal_special_case">this result</a>) – contradiction.


structure_id: B
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would have finite products (by <a href="/category-implication/dual_extensive_assumption">this result</a>) – contradiction.


structure_id: B
property_id: countable coproducts
type: category
proof:
Assume for contradiction that it has countable coproducts. Then it would have countable copowers (by <a href="/category-implication/dual_countable_products_consequence">this result</a>) – contradiction.


structure_id: B
property_id: connected colimits
type: category
proof:
Assume for contradiction that it has connected colimits. Then it would have coequalizers (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>) – contradiction.


structure_id: B
property_id: exact cofiltered limits
type: category
proof:
Assume for contradiction that it has exact cofiltered limits. Then it would be finitely cocomplete (by <a href="/category-implication/dual_exact_filtered_colimits_condition">this result</a>) – contradiction.


structure_id: FI
property_id: connected limits
type: category
proof:
Since it has equalizers and has wide pullbacks, it has connected limits (by <a href="/category-implication/connected_limits_characterization">this result</a>).


structure_id: FI
property_id: quotient-trivial
type: category
proof:
Since it is left cancellative and is mono-regular, it is quotient-trivial (by <a href="/category-implication/dual_subobject-trivial_criterion">this result</a>).


structure_id: FI
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Since it has cofiltered limits and is quotient-trivial, it has cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_filtered_monos_iso">this result</a>).


structure_id: FI
property_id: epi-regular
type: category
proof:
Since it is quotient-trivial, it is epi-regular (by <a href="/category-implication/dual_subobject-trivial_characterization">this result</a>).


structure_id: FI
property_id: multi-cocomplete
type: category
proof:
Since it is locally multi-presentable, it is multi-cocomplete (by <a href="/category-implication/locally_multi-presentable_another_definition">this result</a>).


structure_id: FI
property_id: strict initial object
type: category
proof:
Since it has an initial object and is left cancellative, it has a strict initial object (by <a href="/category-implication/strict_initial_left_criterion">this result</a>).


structure_id: FI
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be connected and would be a groupoid, so it would be strongly connected (by <a href="/category-implication/groupoid_connected">this result</a>) – contradiction.


structure_id: FI
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have binary powers (by <a href="/category-implication/thin_binary_power">this result</a>) – contradiction.


structure_id: FI
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have a generator and would be self-dual, so it would have a cogenerator (by <a href="/category-implication/self-dual_generator">this result</a>) – contradiction.


structure_id: FI
property_id: sifted
type: category
proof:
Assume for contradiction that it is sifted. Then it would be left cancellative and would be sifted, so it would be thin (by <a href="/category-implication/sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


structure_id: FI
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be balanced and would be left cancellative and would be right cancellative, so it would be a groupoid (by <a href="/category-implication/groupoid_criterion">this result</a>) – contradiction.


structure_id: FI
property_id: terminal object
type: category
proof:
Assume for contradiction that it has a terminal object. Then it would have pullbacks and would have a terminal object, so it would have binary products (by <a href="/category-implication/binary_products_criterion">this result</a>) – contradiction.


structure_id: FI
property_id: abelian
type: category
proof:
Assume for contradiction that it is abelian. Then it would have cokernels (by <a href="/category-implication/abelian_definition">this result</a>) – contradiction.


structure_id: FI
property_id: multi-terminal object
type: category
proof:
Assume for contradiction that it has a multi-terminal object. Then it would be connected and would have a multi-terminal object, so it would have a terminal object (by <a href="/category-implication/multi-terminal_with_connected">this result</a>) – contradiction.


structure_id: FI
property_id: distributive
type: category
proof:
Assume for contradiction that it is distributive. Then it would have finite products (by <a href="/category-implication/distributive_assumption">this result</a>) – contradiction.


structure_id: FI
property_id: disjoint finite coproducts
type: category
proof:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would be regular-quotient-trivial, so it would be trivial (by <a href="/category-implication/disjoint_finite_coproducts_disallows_regular-quotient-trivial">this result</a>) – contradiction.


structure_id: FI
property_id: finitely cocomplete
type: category
proof:
Assume for contradiction that it is finitely cocomplete. Then it would have coequalizers (by <a href="/category-implication/dual_finitely_complete_characterization">this result</a>) – contradiction.


structure_id: FI
property_id: codistributive
type: category
proof:
Assume for contradiction that it is codistributive. Then it would have finite products (by <a href="/category-implication/dual_distributive_assumption">this result</a>) – contradiction.


structure_id: FI
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would be finitely complete (by <a href="/category-implication/complete_consequence">this result</a>) – contradiction.


structure_id: FI
property_id: extensive
type: category
proof:
Assume for contradiction that it is extensive. Then it would have disjoint finite coproducts (by <a href="/category-implication/extensive_consequences">this result</a>) – contradiction.


structure_id: FS
property_id: subobject-trivial
type: category
proof:
Since it is epi-regular and is right cancellative, it is subobject-trivial (by <a href="/category-implication/subobject-trivial_criterion">this result</a>).


structure_id: FS
property_id: regular-subobject-trivial
type: category
proof:
Since it is right cancellative, it is regular-subobject-trivial (by <a href="/category-implication/regular-subobject-trivial_criterion">this result</a>).


structure_id: FS
property_id: Cauchy complete
type: category
proof:
Since it is right cancellative, it is Cauchy complete (by <a href="/category-implication/dual_cauchy_complete_criterion">this result</a>).


structure_id: FS
property_id: coreflexive equalizers
type: category
proof:
Since it is right cancellative, it has coreflexive equalizers (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>).


structure_id: FS
property_id: effective cocongruences
type: category
proof:
Since it is right cancellative, it has effective cocongruences (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>).


structure_id: FS
property_id: effective congruences
type: category
proof:
Since it is right cancellative, it has effective congruences (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>).


structure_id: FS
property_id: reflexive coequalizers
type: category
proof:
Since it is right cancellative, it has reflexive coequalizers (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>).


structure_id: FS
property_id: connected colimits
type: category
proof:
Since it has coequalizers and has wide pushouts, it has connected colimits (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>).


structure_id: FS
property_id: filtered-colimit-stable monomorphisms
type: category
proof:
Since it has filtered colimits and is subobject-trivial, it has filtered-colimit-stable monomorphisms (by <a href="/category-implication/filtered_monos_iso">this result</a>).


structure_id: FS
property_id: mono-regular
type: category
proof:
Since it is subobject-trivial, it is mono-regular (by <a href="/category-implication/subobject-trivial_characterization">this result</a>).


structure_id: FS
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would have pullbacks (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


structure_id: FS
property_id: wide pullbacks
type: category
proof:
Assume for contradiction that it has wide pullbacks. Then it would have pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>) – contradiction.


structure_id: FS
property_id: locally finitely multi-presentable
type: category
proof:
Assume for contradiction that it is locally finitely multi-presentable. Then it would be multi-cocomplete (by <a href="/category-implication/locally_finitely_multi-presentable_another_definition">this result</a>) – contradiction.


structure_id: FS
property_id: trivial
type: category
proof:
Assume for contradiction that it is trivial. Then it would be finitary algebraic (by <a href="/category-implication/trivial_consequence">this result</a>) – contradiction.


structure_id: FS
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be balanced and would be left cancellative and would be right cancellative, so it would be a groupoid (by <a href="/category-implication/groupoid_criterion">this result</a>) – contradiction.


structure_id: FS
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would be right cancellative and would have zero morphisms, so it would be thin (by <a href="/category-implication/dual_zero-no-mono">this result</a>) – contradiction.


structure_id: FS
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would be multi-cocomplete (by <a href="/category-implication/dual_multi-complete_generalize_limits">this result</a>) – contradiction.


structure_id: FS
property_id: coproducts
type: category
proof:
Assume for contradiction that it has coproducts. Then it would have coequalizers and would have coproducts, so it would be cocomplete (by <a href="/category-implication/dual_complete_characterization">this result</a>) – contradiction.


structure_id: FinVect_c
property_id: abelian
type: category
proof:
Since it is split abelian, it is abelian (by <a href="/category-implication/split_abelian_condition">this result</a>).


structure_id: FinVect_c
property_id: cogenerator
type: category
proof:
Since it has a generator and is self-dual, it has a cogenerator (by <a href="/category-implication/self-dual_generator">this result</a>).


structure_id: FinVect_c
property_id: coaccessible
type: category
proof:
Since it is accessible and is self-dual, it is coaccessible (by <a href="/category-implication/self-dual_accessible">this result</a>).


structure_id: FinVect_c
property_id: countable powers
type: category
proof:
Assume for contradiction that it has countable powers. Then it would have countable powers and would be essentially countable, so it would be thin (by <a href="/category-implication/freyd_countable">this result</a>) – contradiction.


structure_id: FinVect_c
property_id: countable copowers
type: category
proof:
Assume for contradiction that it has countable copowers. Then it would have countable copowers and would be essentially countable, so it would be thin (by <a href="/category-implication/dual_freyd_countable">this result</a>) – contradiction.


structure_id: FinVect_c
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would be Grothendieck abelian and would be self-dual, so it would be trivial (by <a href="/category-implication/grothendieck_abelian_self-dual">this result</a>) – contradiction.


structure_id: FinVect_c
property_id: sequential colimits
type: category
proof:
Assume for contradiction that it has sequential colimits. Then it would have finite copowers and would have sequential colimits, so it would have countable copowers (by <a href="/category-implication/dual_countable_powers_criterion">this result</a>) – contradiction.


structure_id: FinVect_f
property_id: regular subobject classifier
type: category
proof:
Assume for contradiction that it has a regular subobject classifier. Then it would be Malcev and would have a regular subobject classifier, so it would be regular-subobject-trivial (by <a href="/category-implication/regular_subobject_classifier_disallows_malcev">this result</a>) – contradiction.


structure_id: FinVect_f
property_id: regular quotient object classifier
type: category
proof:
Assume for contradiction that it has a regular quotient object classifier. Then it would be co-Malcev and would have a regular quotient object classifier, so it would be regular-quotient-trivial (by <a href="/category-implication/dual_regular_subobject_classifier_disallows_malcev">this result</a>) – contradiction.


structure_id: FreeAb
property_id: filtered colimits
type: category
proof:
Assume for contradiction that it has filtered colimits. Then it would have ℵ₁-filtered colimits (by <a href="/category-implication/aleph1-filtered-colimits-include-filtered-colimits">this result</a>) – contradiction.


structure_id: FreeAb
property_id: coequalizers
type: category
proof:
Assume for contradiction that it has coequalizers. Then it would have coequalizers and would have countable coproducts, so it would have sequential colimits (by <a href="/category-implication/dual_sequential_limits_criterion">this result</a>) – contradiction.


structure_id: FreeAb
property_id: locally ℵ₁-presentable
type: category
proof:
Assume for contradiction that it is locally ℵ₁-presentable. Then it would be ℵ₁-accessible (by <a href="/category-implication/locally_presentable_definition_countable">this result</a>) – contradiction.


structure_id: FreeAb
property_id: finitely accessible
type: category
proof:
Assume for contradiction that it is finitely accessible. Then it would be ℵ₁-accessible (by <a href="/category-implication/finitely_accessible_raise">this result</a>) – contradiction.


structure_id: FreeAb
property_id: generalized variety
type: category
proof:
Assume for contradiction that it is a generalized variety. Then it would be ℵ₁-accessible (by <a href="/category-implication/generalized_variety_implies_accessible">this result</a>) – contradiction.


structure_id: FreeAb
property_id: pushouts
type: category
proof:
Assume for contradiction that it has pushouts. Then it would have binary coproducts and would have pushouts, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_criterion">this result</a>) – contradiction.


structure_id: FreeAb
property_id: reflexive coequalizers
type: category
proof:
Assume for contradiction that it has reflexive coequalizers. Then it would have binary coproducts and would have reflexive coequalizers, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: FreeAb
property_id: cokernels
type: category
proof:
Assume for contradiction that it has cokernels. Then it would have cokernels and would be preadditive, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_via_kernels">this result</a>) – contradiction.


structure_id: FreeAb
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have coequalizers (by <a href="/category-implication/dual_thin_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: FreeAb
property_id: quotient-trivial
type: category
proof:
Assume for contradiction that it is quotient-trivial. Then it would be epi-regular (by <a href="/category-implication/dual_subobject-trivial_characterization">this result</a>) – contradiction.


structure_id: FreeAb
property_id: multi-algebraic
type: category
proof:
Assume for contradiction that it is multi-algebraic. Then it would be a generalized variety (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>) – contradiction.


structure_id: FreeAb
property_id: quotients of congruences
type: category
proof:
Assume for contradiction that it has quotients of congruences. Then it would be preadditive and would have quotients of congruences and would be regular, so it would have cokernels (by <a href="/category-implication/cokernels_via_congruence_quotients">this result</a>) – contradiction.


structure_id: FreeAb
property_id: one-way
type: category
proof:
Assume for contradiction that it is one-way. Then it would have reflexive coequalizers (by <a href="/category-implication/one-way_reflexive">this result</a>) – contradiction.


structure_id: FreeAb
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would have copowers and would be essentially small, so it would be thin (by <a href="/category-implication/dual_freyd_small">this result</a>) – contradiction.


structure_id: Grp
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have biproducts and would have filtered colimits and would have filtered-colimit-stable monomorphisms and would have products, so it would satisfy CIP (by <a href="/category-implication/CIP_criterion">this result</a>) – contradiction.


structure_id: Grp
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have cartesian filtered colimits and would be self-dual, so it would have cocartesian cofiltered limits (by <a href="/category-implication/self-dual_cartesian filtered colimits">this result</a>) – contradiction.


structure_id: Haus
property_id: coproducts
type: category
proof:
Since it is infinitary extensive, it has coproducts (by <a href="/category-implication/infinitary_extensive_assumption">this result</a>).


structure_id: Haus
property_id: extensive
type: category
proof:
Since it is infinitary extensive, it is extensive (by <a href="/category-implication/infinitary_extensive_finitary">this result</a>).


structure_id: Haus
property_id: finitely cocomplete
type: category
proof:
Since it is co-Malcev, it is finitely cocomplete (by <a href="/category-implication/dual_malcev_condition">this result</a>).


structure_id: Haus
property_id: infinitary distributive
type: category
proof:
Since it has finite products and is infinitary extensive, it is infinitary distributive (by <a href="/category-implication/infinitary_extensive_distributivity">this result</a>).


structure_id: Haus
property_id: countably distributive
type: category
proof:
Since it is infinitary distributive, it is countably distributive (by <a href="/category-implication/infinitary_distributive_trivial">this result</a>).


structure_id: Haus
property_id: locally presentable
type: category
proof:
Assume for contradiction that it is locally presentable. Then it would be accessible (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


structure_id: Haus
property_id: ℵ₁-accessible
type: category
proof:
Assume for contradiction that it is ℵ₁-accessible. Then it would be accessible (by <a href="/category-implication/countably_accessible_special_case">this result</a>) – contradiction.


structure_id: Haus
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would be Cauchy complete and would be essentially small, so it would be accessible (by <a href="/category-implication/small_accessible_characterization">this result</a>) – contradiction.


structure_id: Haus
property_id: abelian
type: category
proof:
Assume for contradiction that it is abelian. Then it would be regular (by <a href="/category-implication/abelian_implies_regular">this result</a>) – contradiction.


structure_id: Haus
property_id: locally strongly finitely presentable
type: category
proof:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be regular (by <a href="/category-implication/many_sorted_algebraic_implies_regular">this result</a>) – contradiction.


structure_id: Haus
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have filtered colimits, so it would have cartesian filtered colimits (by <a href="/category-implication/ccc_cartesian_filtered_colimits">this result</a>) – contradiction.


structure_id: Haus
property_id: locally cartesian closed
type: category
proof:
Assume for contradiction that it is locally cartesian closed. Then it would have coequalizers and would be finitely complete and would be locally cartesian closed, so it would be regular (by <a href="/category-implication/lcc_implies_regular">this result</a>) – contradiction.


structure_id: Haus
property_id: coextensive
type: category
proof:
Assume for contradiction that it is coextensive. Then it would be coextensive and would have filtered colimits and would have an initial object, so it would have cartesian filtered colimits (by <a href="/category-implication/dual_extensive_cocartesian_cofiltered_limits">this result</a>) – contradiction.


structure_id: Haus
property_id: generalized variety
type: category
proof:
Assume for contradiction that it is a generalized variety. Then it would be cocomplete and would be a generalized variety, so it would be locally strongly finitely presentable (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>) – contradiction.


structure_id: Haus
property_id: effective congruences
type: category
proof:
Assume for contradiction that it has effective congruences. Then it would have effective congruences and would be extensive, so it would be mono-regular (by <a href="/category-implication/pretopos_balanced">this result</a>) – contradiction.


structure_id: J2
property_id: coproducts
type: category
proof:
Since it is a Grothendieck topos, it has coproducts (by <a href="/category-implication/grothendieck_topos_definition">this result</a>).


structure_id: J2
property_id: elementary topos
type: category
proof:
Since it is a Grothendieck topos, it is an elementary topos (by <a href="/category-implication/grothendieck_topos_definition">this result</a>).


structure_id: J2
property_id: generating set
type: category
proof:
Since it is a Grothendieck topos, it has a generating set (by <a href="/category-implication/grothendieck_topos_definition">this result</a>).


structure_id: J2
property_id: cogenerator
type: category
proof:
Since it is a Grothendieck topos, it has a cogenerator (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


structure_id: J2
property_id: exact filtered colimits
type: category
proof:
Since it is a Grothendieck topos, it has exact filtered colimits (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


structure_id: J2
property_id: infinitary extensive
type: category
proof:
Since it is a Grothendieck topos, it is infinitary extensive (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


structure_id: J2
property_id: locally presentable
type: category
proof:
Since it is a Grothendieck topos, it is locally presentable (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>).


structure_id: J2
property_id: powers
type: category
proof:
Since it is cartesian closed and has copowers, it has powers (by <a href="/category-implication/power_construction">this result</a>).


structure_id: J2
property_id: infinitary distributive
type: category
proof:
Since it is cartesian closed and has coproducts, it is infinitary distributive (by <a href="/category-implication/infinitary_distributive_criterion">this result</a>).


structure_id: J2
property_id: ℵ₂-small powers
type: category
proof:
Since it has powers, it has ℵ₂-small powers (by <a href="/category-implication/powers_include_aleph2-small_powers">this result</a>).


structure_id: J2
property_id: countable powers
type: category
proof:
Since it is cartesian closed and has countable copowers, it has countable powers (by <a href="/category-implication/countable_power_construction">this result</a>).


structure_id: J2
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be preadditive (by <a href="/category-implication/additive_definition">this result</a>) – contradiction.


structure_id: J2
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have disjoint finite coproducts and would be thin, so it would be trivial (by <a href="/category-implication/disjoint_coproducts_thin_trivial">this result</a>) – contradiction.


structure_id: J2
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Assume for contradiction that it has cofiltered-limit-stable epimorphisms. Then it would have cofiltered-limit-stable epimorphisms and would have countable coproducts and would be an elementary topos, so it would be trivial (by <a href="/category-implication/topos_no_stable_epis">this result</a>) – contradiction.


structure_id: J2
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be regular-subobject-trivial (by <a href="/category-implication/regular-subobject-trivial_criterion">this result</a>) – contradiction.


structure_id: R-Mod_div
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be left cancellative and would be mono-regular, so it would be quotient-trivial (by <a href="/category-implication/dual_subobject-trivial_criterion">this result</a>) – contradiction.


structure_id: LRS_R
property_id: accessible
type: category
proof:
Assume for contradiction that it is accessible. Then it would have a generating set (by <a href="/category-implication/accessible_trivial_consequence">this result</a>) – contradiction.


structure_id: LRS_R
property_id: exact filtered colimits
type: category
proof:
Assume for contradiction that it has exact filtered colimits. Then it would have cartesian filtered colimits (by <a href="/category-implication/exact_includes_cartesian_filtered_colimits">this result</a>) – contradiction.


structure_id: LRS_R
property_id: generator
type: category
proof:
Assume for contradiction that it has a generator. Then it would have a generating set (by <a href="/category-implication/generator_consequence">this result</a>) – contradiction.


structure_id: LRS_R
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would have a generating set (by <a href="/category-implication/essentially_small_consequences">this result</a>) – contradiction.


structure_id: LRS_R
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would have a generating set (by <a href="/category-implication/grothendieck_topos_definition">this result</a>) – contradiction.


structure_id: LRS_R
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have cocartesian cofiltered limits and would be self-dual, so it would have cartesian filtered colimits (by <a href="/category-implication/self-dual_cocartesian cofiltered limits">this result</a>) – contradiction.


structure_id: LRS_R
property_id: finitely accessible
type: category
proof:
Assume for contradiction that it is finitely accessible. Then it would be cocomplete and would be finitely accessible, so it would be locally finitely presentable (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


structure_id: Meas
property_id: filtered colimits
type: category
proof:
Since it has filtered-colimit-stable monomorphisms, it has filtered colimits (by <a href="/category-implication/filtered_monos_assumption">this result</a>).


structure_id: Meas
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be finitely complete and would be thin, so it would be regular (by <a href="/category-implication/thin_implies_regular">this result</a>) – contradiction.


structure_id: Meas
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would have biproducts (by <a href="/category-implication/additive_biproducts">this result</a>) – contradiction.


structure_id: Met_c
property_id: finite products
type: category
proof:
Since it has countable products, it has finite products (by <a href="/category-implication/countable_products_consequence">this result</a>).


structure_id: Met_c
property_id: finitely complete
type: category
proof:
Since it has equalizers and has finite products, it is finitely complete (by <a href="/category-implication/finitely_complete_characterization">this result</a>).


structure_id: Met_c
property_id: reflexive coequalizers
type: category
proof:
Assume for contradiction that it has reflexive coequalizers. Then it would have quotients of congruences (by <a href="/category-implication/congruence_quotients_are_reflexive_coequalizers">this result</a>) – contradiction.


structure_id: Met_c
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would have quotients of congruences (by <a href="/category-implication/core-hin_quotients">this result</a>) – contradiction.


structure_id: Met_c
property_id: ℵ₂-small products
type: category
proof:
Assume for contradiction that it has ℵ₂-small products. Then it would have ℵ₂-small powers (by <a href="/category-implication/aleph2-small_products_consequences">this result</a>) – contradiction.


structure_id: Met_c
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be self-dual and would have ℵ₂-small copowers, so it would have ℵ₂-small powers (by <a href="/category-implication/self-dual_ℵ₂-small copowers">this result</a>) – contradiction.


structure_id: Met_c
property_id: cartesian closed
type: category
proof:
Assume for contradiction that it is cartesian closed. Then it would be cartesian closed and would have copowers, so it would have powers (by <a href="/category-implication/power_construction">this result</a>) – contradiction.


structure_id: Met_c
property_id: subobject-trivial
type: category
proof:
Assume for contradiction that it is subobject-trivial. Then it would have reflexive coequalizers (by <a href="/category-implication/reflexive_pair_trivial_2">this result</a>) – contradiction.


structure_id: Met_c
property_id: quotient-trivial
type: category
proof:
Assume for contradiction that it is quotient-trivial. Then it would have reflexive coequalizers (by <a href="/category-implication/dual_reflexive_pair_trivial_2">this result</a>) – contradiction.


structure_id: Met_c
property_id: ℵ₁-cofiltered limits
type: category
proof:
Assume for contradiction that it has ℵ₁-cofiltered limits. Then it would have countable powers and would have ℵ₁-cofiltered limits, so it would have powers (by <a href="/category-implication/dual_copowers_criterion_countable">this result</a>) – contradiction.


structure_id: Met
property_id: sifted colimits
type: category
proof:
Since it has filtered colimits and has pullbacks and has reflexive coequalizers, it has sifted colimits (by <a href="/category-implication/sifted_colimits_criterion">this result</a>).


structure_id: Met
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would be finitely cocomplete (by <a href="/category-implication/dual_complete_consequence">this result</a>) – contradiction.


structure_id: Met_oo
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have disjoint finite coproducts and would be self-dual, so it would have disjoint finite products (by <a href="/category-implication/self-dual_disjoint finite coproducts">this result</a>) – contradiction.


structure_id: Mon
property_id: finitely complete
type: category
proof:
Since it is unital, it is finitely complete (by <a href="/category-implication/unital_assumptions">this result</a>).


structure_id: Setne
property_id: finite products
type: category
proof:
Since it has a natural numbers object, it has finite products (by <a href="/category-implication/nno_assumption">this result</a>).


structure_id: Setne
property_id: cartesian filtered colimits
type: category
proof:
Since it is cartesian closed and has filtered colimits, it has cartesian filtered colimits (by <a href="/category-implication/ccc_cartesian_filtered_colimits">this result</a>).


structure_id: Setne
property_id: coequalizers
type: category
proof:
Since it has binary coproducts and has reflexive coequalizers, it has coequalizers (by <a href="/category-implication/dual_equalizers_via_coreflexive_equalizers">this result</a>).


structure_id: Setne
property_id: equalizers
type: category
proof:
Assume for contradiction that it has equalizers. Then it would have countable products and would have equalizers, so it would have sequential limits (by <a href="/category-implication/sequential_limits_criterion">this result</a>) – contradiction.


structure_id: Setne
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would be Cauchy complete and would be essentially small, so it would be coaccessible (by <a href="/category-implication/dual_small_accessible_characterization">this result</a>) – contradiction.


structure_id: Setne
property_id: coreflexive equalizers
type: category
proof:
Assume for contradiction that it has coreflexive equalizers. Then it would have coquotients of cocongruences (by <a href="/category-implication/dual_congruence_quotients_are_reflexive_coequalizers">this result</a>) – contradiction.


structure_id: Setne
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would have coquotients of cocongruences (by <a href="/category-implication/dual_core-hin_quotients">this result</a>) – contradiction.


structure_id: Setne
property_id: finitely complete
type: category
proof:
Assume for contradiction that it is finitely complete. Then it would be cofiltered (by <a href="/category-implication/dual_filtered_criterion">this result</a>) – contradiction.


structure_id: Setne
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be accessible and would be self-dual, so it would be coaccessible (by <a href="/category-implication/self-dual_accessible">this result</a>) – contradiction.


structure_id: Setne
property_id: unital
type: category
proof:
Assume for contradiction that it is unital. Then it would be finitely complete (by <a href="/category-implication/unital_assumptions">this result</a>) – contradiction.


structure_id: Setne
property_id: pullbacks
type: category
proof:
Assume for contradiction that it has pullbacks. Then it would have binary products and would have pullbacks, so it would have equalizers (by <a href="/category-implication/equalizers_criterion">this result</a>) – contradiction.


structure_id: Setne
property_id: subobject-trivial
type: category
proof:
Assume for contradiction that it is subobject-trivial. Then it would have coreflexive equalizers (by <a href="/category-implication/reflexive_pair_trivial_2">this result</a>) – contradiction.


structure_id: Setne
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have equalizers (by <a href="/category-implication/thin_regular-subobject-trivial">this result</a>) – contradiction.


structure_id: Setne
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would be finitely complete (by <a href="/category-implication/topos_definition">this result</a>) – contradiction.


structure_id: Setne
property_id: quasitopos
type: category
proof:
Assume for contradiction that it is a quasitopos. Then it would be finitely complete (by <a href="/category-implication/quasitopos_definition">this result</a>) – contradiction.


structure_id: Setne
property_id: quotient-trivial
type: category
proof:
Assume for contradiction that it is quotient-trivial. Then it would have coreflexive equalizers (by <a href="/category-implication/dual_reflexive_pair_trivial_2">this result</a>) – contradiction.


structure_id: Setne
property_id: initial object
type: category
proof:
Assume for contradiction that it has an initial object. Then it would be ℵ₁-cofiltered (by <a href="/category-implication/dual_terminal_object_yields_kappa_filtered">this result</a>) – contradiction.


structure_id: Setne
property_id: strict terminal object
type: category
proof:
Assume for contradiction that it has a strict terminal object. Then it would have a natural numbers object and would have a strict terminal object, so it would be one-way (by <a href="/category-implication/nno_terminal">this result</a>) – contradiction.


structure_id: Setne
property_id: regular-subobject-trivial
type: category
proof:
Assume for contradiction that it is regular-subobject-trivial. Then it would be mono-regular and would be regular-subobject-trivial, so it would be subobject-trivial (by <a href="/category-implication/subobject-trivial_characterization">this result</a>) – contradiction.


structure_id: Setne
property_id: multi-initial object
type: category
proof:
Assume for contradiction that it has a multi-initial object. Then it would be connected and would have a multi-initial object, so it would have an initial object (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>) – contradiction.


structure_id: Setne
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would be cocomplete and would be finitely accessible, so it would be locally finitely presentable (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>) – contradiction.


structure_id: Setne
property_id: multi-cocomplete
type: category
proof:
Assume for contradiction that it is multi-cocomplete. Then it would be accessible and would be multi-cocomplete, so it would be locally multi-presentable (by <a href="/category-implication/locally_multi-presentable_another_definition">this result</a>) – contradiction.


structure_id: Setne
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would have finite products and would be preadditive, so it would have finite coproducts (by <a href="/category-implication/dual_preadditive_products_criterion">this result</a>) – contradiction.


structure_id: Setne
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would have finite coproducts (by <a href="/category-implication/dual_additive_definition">this result</a>) – contradiction.


structure_id: Setne
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would have a terminal object and would have zero morphisms, so it would be pointed (by <a href="/category-implication/dual_pointed_characterization">this result</a>) – contradiction.


structure_id: Setne
property_id: locally cocartesian coclosed
type: category
proof:
Assume for contradiction that it is locally cocartesian coclosed. Then it would have disjoint finite products and would be locally cocartesian coclosed, so it would be coextensive (by <a href="/category-implication/dual_lcc_implies_extensive">this result</a>) – contradiction.


structure_id: SetxSet
property_id: well-copowered
type: category
proof:
Since it is an elementary topos and is locally essentially small, it is well-copowered (by <a href="/category-implication/topos_well-copowered_criterion">this result</a>).


structure_id: SetxSet
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would have coproducts and would have a generating set and would have zero morphisms, so it would have a generator (by <a href="/category-implication/generator_via_coproduct">this result</a>) – contradiction.


structure_id: SetxSet
property_id: disjoint finite products
type: category
proof:
Assume for contradiction that it has disjoint finite products. Then it would have disjoint finite products and would have products, so it would have disjoint products (by <a href="/category-implication/dual_disjoint_coproducts_equivalence">this result</a>) – contradiction.


structure_id: Set_*
property_id: finitely complete
type: category
proof:
Since it has a subobject classifier, it is finitely complete (by <a href="/category-implication/subobject_classifier_consequence">this result</a>).


structure_id: Set_*
property_id: normal
type: category
proof:
Since it is pointed and has a subobject classifier, it is normal (by <a href="/category-implication/subobject_classifier_pointed_case">this result</a>).


structure_id: Set_*
property_id: effective cocongruences
type: category
proof:
Since it is Barr-coexact, it has effective cocongruences (by <a href="/category-implication/dual_Barr-exact_definition">this result</a>).


structure_id: Set_*
property_id: well-powered
type: category
proof:
Since it is locally essentially small and has a subobject classifier, it is well-powered (by <a href="/category-implication/subobject_classifier_well-powered">this result</a>).


structure_id: Set_*
property_id: coquotients of cocongruences
type: category
proof:
Since it is coregular and has effective cocongruences, it has coquotients of cocongruences (by <a href="/category-implication/dual_regular_effective_congruences_implies_quotients">this result</a>).


structure_id: Set_*
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be additive and would have effective cocongruences, so it would be conormal (by <a href="/category-implication/dual_additive_effective_congruences_imply_normal">this result</a>) – contradiction.


structure_id: Set_*
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would be epi-regular and would be preadditive, so it would be conormal (by <a href="/category-implication/dual_normal_criterion">this result</a>) – contradiction.


structure_id: Set_*
property_id: quotient object classifier
type: category
proof:
Assume for contradiction that it has a quotient object classifier. Then it would be pointed and would have a quotient object classifier, so it would be conormal (by <a href="/category-implication/dual_subobject_classifier_pointed_case">this result</a>) – contradiction.


structure_id: Set_*
property_id: regular quotient object classifier
type: category
proof:
Assume for contradiction that it has a regular quotient object classifier. Then it would be epi-regular and would have a regular quotient object classifier, so it would have a quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_suffice">this result</a>) – contradiction.


structure_id: Set_*
property_id: regular-quotient-trivial
type: category
proof:
Assume for contradiction that it is regular-quotient-trivial. Then it would be epi-regular and would be regular-quotient-trivial, so it would be quotient-trivial (by <a href="/category-implication/dual_subobject-trivial_characterization">this result</a>) – contradiction.


structure_id: Top_*
property_id: finitely cocomplete
type: category
proof:
Since it is counital, it is finitely cocomplete (by <a href="/category-implication/dual_unital_assumptions">this result</a>).


structure_id: Top_*
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would have effective congruences (by <a href="/category-implication/topos_consequence">this result</a>) – contradiction.


structure_id: Top_*
property_id: Grothendieck abelian
type: category
proof:
Assume for contradiction that it is Grothendieck abelian. Then it would be locally presentable (by <a href="/category-implication/grothendieck_abelian_presentable">this result</a>) – contradiction.


structure_id: Pos
property_id: cocomplete
type: category
proof:
Since it is locally finitely presentable, it is cocomplete (by <a href="/category-implication/locally_presentable_definition_finite">this result</a>).


structure_id: Pos
property_id: locally finitely multi-presentable
type: category
proof:
Since it is finitely accessible and is multi-cocomplete, it is locally finitely multi-presentable (by <a href="/category-implication/locally_finitely_multi-presentable_another_definition">this result</a>).


structure_id: Pos
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be coregular and would be self-dual, so it would be regular (by <a href="/category-implication/self-dual_coregular">this result</a>) – contradiction.


structure_id: Prost
property_id: finitely complete
type: category
proof:
Since it has a regular subobject classifier, it is finitely complete (by <a href="/category-implication/regular_subobject_assumption">this result</a>).


structure_id: Prost
property_id: subobject classifier
type: category
proof:
Assume for contradiction that it has a subobject classifier. Then it would be cartesian closed and would be finitely complete and would have a subobject classifier, so it would be an elementary topos (by <a href="/category-implication/topos_definition">this result</a>) – contradiction.


structure_id: Prost
property_id: mono-regular
type: category
proof:
Assume for contradiction that it is mono-regular. Then it would be mono-regular and would have a regular subobject classifier, so it would have a subobject classifier (by <a href="/category-implication/regular_subobjects_suffice">this result</a>) – contradiction.


structure_id: Prost
property_id: balanced
type: category
proof:
Assume for contradiction that it is balanced. Then it would be balanced and would be coregular, so it would be mono-regular (by <a href="/category-implication/dual_regular_balanced_epi-regular">this result</a>) – contradiction.


structure_id: PMet
property_id: finite products
type: category
proof:
Since it has binary products and has a terminal object, it has finite products (by <a href="/category-implication/finite_products_characterization">this result</a>).


structure_id: Sch_R
property_id: binary products
type: category
proof:
Since it has pullbacks and has a terminal object, it has binary products (by <a href="/category-implication/binary_products_criterion">this result</a>).


structure_id: Sch_R
property_id: equalizers
type: category
proof:
Since it has binary products and has pullbacks, it has equalizers (by <a href="/category-implication/equalizers_criterion">this result</a>).


structure_id: Sch_R
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have coquotients of cocongruences and would be self-dual, so it would have quotients of congruences (by <a href="/category-implication/self-dual_coquotients of cocongruences">this result</a>) – contradiction.


structure_id: Sch_R
property_id: sifted colimits
type: category
proof:
Assume for contradiction that it has sifted colimits. Then it would have reflexive coequalizers (by <a href="/category-implication/sifted_colimits_consequence">this result</a>) – contradiction.


structure_id: SemiGrp
property_id: coproducts
type: category
proof:
Since it has disjoint coproducts, it has coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>).


structure_id: SemiGrp
property_id: disjoint finite coproducts
type: category
proof:
Since it has disjoint coproducts, it has disjoint finite coproducts (by <a href="/category-implication/disjoint_coproducts_equivalence">this result</a>).


structure_id: SemiGrp
property_id: connected
type: category
proof:
Since it has an initial object, it is connected (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>).


structure_id: SemiGrp
property_id: products
type: category
proof:
Since it has cofiltered limits and has finite products, it has products (by <a href="/category-implication/products_criterion">this result</a>).


structure_id: SemiGrp
property_id: countable products
type: category
proof:
Since it has finite products and has sequential limits, it has countable products (by <a href="/category-implication/countable_products_criterion">this result</a>).


structure_id: SemiGrp
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have effective congruences and would be self-dual, so it would have effective cocongruences (by <a href="/category-implication/self-dual_effective congruences">this result</a>) – contradiction.


structure_id: SemiGrp
property_id: distributive
type: category
proof:
Assume for contradiction that it is distributive. Then it would have cartesian filtered colimits and would have countable coproducts and would be distributive, so it would be countably distributive (by <a href="/category-implication/countably_distributive_filtered_criterion">this result</a>) – contradiction.


structure_id: Rel
property_id: CSP
type: category
proof:
Since it satisfies CIP and is self-dual, it satisfies CSP (by <a href="/category-implication/self-dual_CIP">this result</a>).


structure_id: Rel
property_id: effective cocongruences
type: category
proof:
Since it has effective congruences and is self-dual, it has effective cocongruences (by <a href="/category-implication/self-dual_effective congruences">this result</a>).


structure_id: Rel
property_id: cokernels
type: category
proof:
Since it has kernels and is self-dual, it has cokernels (by <a href="/category-implication/self-dual_kernels">this result</a>).


structure_id: Rel
property_id: epi-regular
type: category
proof:
Since it is mono-regular and is self-dual, it is epi-regular (by <a href="/category-implication/self-dual_mono-regular">this result</a>).


structure_id: Rel
property_id: coquotients of cocongruences
type: category
proof:
Since it has quotients of congruences and is self-dual, it has coquotients of cocongruences (by <a href="/category-implication/self-dual_quotients of congruences">this result</a>).


structure_id: Rel
property_id: well-copowered
type: category
proof:
Since it is self-dual and is well-powered, it is well-copowered (by <a href="/category-implication/self-dual_well-powered">this result</a>).


structure_id: Rel
property_id: accessible
type: category
proof:
Assume for contradiction that it is accessible. Then it would be Cauchy complete (by <a href="/category-implication/accessible_require_Cauchy_complete">this result</a>) – contradiction.


structure_id: Rel
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be Cauchy complete (by <a href="/category-implication/cauchy_complete_criterion">this result</a>) – contradiction.


structure_id: Rel
property_id: equalizers
type: category
proof:
Assume for contradiction that it has equalizers. Then it would be Cauchy complete (by <a href="/category-implication/equalizers_consequence">this result</a>) – contradiction.


structure_id: Rel
property_id: sequential colimits
type: category
proof:
Assume for contradiction that it has sequential colimits. Then it would be Cauchy complete (by <a href="/category-implication/sequential_colimits_consequence">this result</a>) – contradiction.


structure_id: Rel
property_id: ℵ₁-filtered colimits
type: category
proof:
Assume for contradiction that it has ℵ₁-filtered colimits. Then it would be Cauchy complete (by <a href="/category-implication/kappa-filtered-colimits_require_Cauchy_complete">this result</a>) – contradiction.


structure_id: Rel
property_id: coaccessible
type: category
proof:
Assume for contradiction that it is coaccessible. Then it would be Cauchy complete (by <a href="/category-implication/dual_accessible_require_Cauchy_complete">this result</a>) – contradiction.


structure_id: Rel
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be Cauchy complete (by <a href="/category-implication/dual_cauchy_complete_criterion">this result</a>) – contradiction.


structure_id: Rel
property_id: coequalizers
type: category
proof:
Assume for contradiction that it has coequalizers. Then it would be Cauchy complete (by <a href="/category-implication/dual_equalizers_consequence">this result</a>) – contradiction.


structure_id: Rel
property_id: sequential limits
type: category
proof:
Assume for contradiction that it has sequential limits. Then it would be Cauchy complete (by <a href="/category-implication/dual_sequential_colimits_consequence">this result</a>) – contradiction.


structure_id: Rel
property_id: ℵ₁-cofiltered limits
type: category
proof:
Assume for contradiction that it has ℵ₁-cofiltered limits. Then it would be Cauchy complete (by <a href="/category-implication/dual_kappa-filtered-colimits_require_Cauchy_complete">this result</a>) – contradiction.


structure_id: Rel
property_id: conormal
type: category
proof:
Assume for contradiction that it is conormal. Then it would be conormal and would be self-dual, so it would be normal (by <a href="/category-implication/self-dual_conormal">this result</a>) – contradiction.


structure_id: Rel
property_id: coreflexive equalizers
type: category
proof:
Assume for contradiction that it has coreflexive equalizers. Then it would have binary products and would have coreflexive equalizers, so it would have equalizers (by <a href="/category-implication/equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: Mono
property_id: infinitary extensive
type: category
proof:
Since it is cocomplete and is extensive and is locally cartesian closed, it is infinitary extensive (by <a href="/category-implication/lcc_extensive_yields_infinitary">this result</a>).


structure_id: Mono
property_id: cartesian closed
type: category
proof:
Since it is locally cartesian closed and has a terminal object, it is cartesian closed (by <a href="/category-implication/locally_cartesian_closed_with_terminal_is_closed">this result</a>).


structure_id: Mono
property_id: infinitary distributive
type: category
proof:
Since it has cartesian filtered colimits and has coproducts and is distributive, it is infinitary distributive (by <a href="/category-implication/infinite_distributive_filtered_criterion">this result</a>).


structure_id: Mono
property_id: countably distributive
type: category
proof:
Since it has cartesian filtered colimits and has countable coproducts and is distributive, it is countably distributive (by <a href="/category-implication/countably_distributive_filtered_criterion">this result</a>).


structure_id: Mono
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would have cocartesian cofiltered limits and would be thin, so it would have exact cofiltered limits (by <a href="/category-implication/dual_thin_exact_filtered_colimits">this result</a>) – contradiction.


structure_id: Set_f
property_id: coequalizers
type: category
proof:
Assume for contradiction that it has coequalizers. Then it would have coequalizers and would be semi-strongly connected, so it would be filtered (by <a href="/category-implication/filtered_via_coequalizers">this result</a>) – contradiction.


structure_id: Set_f
property_id: terminal object
type: category
proof:
Assume for contradiction that it has a terminal object. Then it would be ℵ₁-filtered (by <a href="/category-implication/terminal_object_yields_kappa_filtered">this result</a>) – contradiction.


structure_id: Set_f
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would be finitely cocomplete (by <a href="/category-implication/topos_consequence">this result</a>) – contradiction.


structure_id: Set_f
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would be cosifted and would be right cancellative, so it would be thin (by <a href="/category-implication/dual_sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


structure_id: Set_f
property_id: core-thin
type: category
proof:
Assume for contradiction that it is core-thin. Then it would have binary copowers and would be core-thin, so it would be thin (by <a href="/category-implication/dual_core-thin_products_thin">this result</a>) – contradiction.


structure_id: Set_f
property_id: regular
type: category
proof:
Assume for contradiction that it is regular. Then it would be epi-regular and would be extensive and would be regular, so it would be co-Malcev (by <a href="/category-implication/regular_epi-regular_extensive_consequences">this result</a>) – contradiction.


structure_id: Set_f
property_id: products
type: category
proof:
Assume for contradiction that it has products. Then it would have ℵ₂-small products (by <a href="/category-implication/products_include_aleph2-small_products">this result</a>) – contradiction.


structure_id: Set_f
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would be accessible and would be complete, so it would be locally presentable (by <a href="/category-implication/locally_presentable_another_definition">this result</a>) – contradiction.


structure_id: Man
property_id: equalizers
type: category
proof:
Assume for contradiction that it has equalizers. Then it would have binary products and would have equalizers, so it would have pullbacks (by <a href="/category-implication/pullbacks_criterion">this result</a>) – contradiction.


structure_id: Man
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would have reflexive coequalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>) – contradiction.


structure_id: Man
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be semi-strongly connected and would be thin, so it would be locally cartesian closed (by <a href="/category-implication/sequential_implies_lcc">this result</a>) – contradiction.


structure_id: Man
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would have reflexive coequalizers (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>) – contradiction.


structure_id: TorsAb
property_id: abelian
type: category
proof:
Since it is additive and has cokernels and is conormal and has kernels and is normal, it is abelian (by <a href="/category-implication/abelian_definition">this result</a>).


structure_id: TorsAb
property_id: pointed
type: category
proof:
Since it has an initial object and has zero morphisms, it is pointed (by <a href="/category-implication/pointed_characterization">this result</a>).


structure_id: TorsAb
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would be finitely accessible and would be left cancellative, so it would be a generalized variety (by <a href="/category-implication/generalized_variety_left_cancellative">this result</a>) – contradiction.


structure_id: Z
property_id: accessible
type: category
proof:
Assume for contradiction that it is accessible. Then it would be well-powered (by <a href="/category-implication/accessible_well-powered">this result</a>) – contradiction.


structure_id: Z
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would be locally essentially small (by <a href="/category-implication/preadditive_consequence">this result</a>) – contradiction.


structure_id: Z
property_id: essentially discrete
type: category
proof:
Assume for contradiction that it is essentially discrete. Then it would be locally essentially small (by <a href="/category-implication/essentially_discrete_consequence">this result</a>) – contradiction.


structure_id: Z
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be well-powered (by <a href="/category-implication/groupoid_consequence">this result</a>) – contradiction.


structure_id: Z
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would be locally essentially small (by <a href="/category-implication/essentially_small_consequences">this result</a>) – contradiction.


structure_id: Z
property_id: locally small
type: category
proof:
Assume for contradiction that it is locally small. Then it would be locally essentially small (by <a href="/category-implication/locally_small_consequence">this result</a>) – contradiction.


structure_id: Z
property_id: locally finite
type: category
proof:
Assume for contradiction that it is locally finite. Then it would be locally essentially small (by <a href="/category-implication/locally_finite_consequence">this result</a>) – contradiction.


structure_id: Z
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would be locally essentially small (by <a href="/category-implication/grothendieck_topos_definition">this result</a>) – contradiction.


structure_id: Z
property_id: coaccessible
type: category
proof:
Assume for contradiction that it is coaccessible. Then it would be locally essentially small (by <a href="/category-implication/dual_accessible_locally_small">this result</a>) – contradiction.


structure_id: Z
property_id: well-copowered
type: category
proof:
Assume for contradiction that it is well-copowered. Then it would be coregular and would be mono-regular and would be well-copowered, so it would be well-powered (by <a href="/category-implication/dual_regular_well-powered_well-copowered">this result</a>) – contradiction.


structure_id: Z
property_id: finitary algebraic
type: category
proof:
Assume for contradiction that it is finitary algebraic. Then it would be well-copowered (by <a href="/category-implication/algebraic_well-copowered">this result</a>) – contradiction.


structure_id: Z
property_id: locally strongly finitely presentable
type: category
proof:
Assume for contradiction that it is locally strongly finitely presentable. Then it would be a generalized variety (by <a href="/category-implication/locally_strongly_finitely_presentable_definition">this result</a>) – contradiction.


structure_id: BG_f
property_id: strongly connected
type: category
proof:
Since it is connected and is a groupoid, it is strongly connected (by <a href="/category-implication/groupoid_connected">this result</a>).


structure_id: BG_f
property_id: essentially small
type: category
proof:
Since it is small, it is essentially small (by <a href="/category-implication/small_consequence">this result</a>).


structure_id: BG_f
property_id: locally small
type: category
proof:
Since it is small, it is locally small (by <a href="/category-implication/small_consequence">this result</a>).


structure_id: BG_f
property_id: countable
type: category
proof:
Since it is finite, it is countable (by <a href="/category-implication/finite_consequence">this result</a>).


structure_id: BG_f
property_id: essentially finite
type: category
proof:
Since it is finite, it is essentially finite (by <a href="/category-implication/finite_consequence">this result</a>).


structure_id: BG_f
property_id: wide pullbacks
type: category
proof:
Since it is essentially finite and has pullbacks, it has wide pullbacks (by <a href="/category-implication/wide_pullbacks_finite">this result</a>).


structure_id: BG_f
property_id: essentially countable
type: category
proof:
Since it is essentially finite, it is essentially countable (by <a href="/category-implication/essentially_finite_raise">this result</a>).


structure_id: BG_f
property_id: locally finite
type: category
proof:
Since it is essentially finite, it is locally finite (by <a href="/category-implication/essentially_finite_raise">this result</a>).


structure_id: BG_f
property_id: wide pushouts
type: category
proof:
Since it is essentially finite and has pushouts, it has wide pushouts (by <a href="/category-implication/dual_wide_pullbacks_finite">this result</a>).


structure_id: BG_f
property_id: discrete
type: category
proof:
Assume for contradiction that it is discrete. Then it would be essentially discrete (by <a href="/category-implication/discrete_consequences">this result</a>) – contradiction.


structure_id: BG_f
property_id: finite products
type: category
proof:
Assume for contradiction that it has finite products. Then it would have finite products and would be strongly connected, so it would have disjoint finite products (by <a href="/category-implication/strongly_connected_disjoint_products">this result</a>) – contradiction.


structure_id: BG_f
property_id: finite powers
type: category
proof:
Assume for contradiction that it has finite powers. Then it would be essentially finite and would have finite powers, so it would be thin (by <a href="/category-implication/freyd_finite">this result</a>) – contradiction.


structure_id: BG_f
property_id: finite coproducts
type: category
proof:
Assume for contradiction that it has finite coproducts. Then it would have finite coproducts and would be strongly connected, so it would have disjoint finite coproducts (by <a href="/category-implication/dual_strongly_connected_disjoint_products">this result</a>) – contradiction.


structure_id: BG_f
property_id: cosifted
type: category
proof:
Assume for contradiction that it is cosifted. Then it would be cosifted and would be right cancellative, so it would be thin (by <a href="/category-implication/dual_sifted_left_cancellative_implies_thin">this result</a>) – contradiction.


structure_id: BG_f
property_id: finite copowers
type: category
proof:
Assume for contradiction that it has finite copowers. Then it would be essentially finite and would have finite copowers, so it would be thin (by <a href="/category-implication/dual_freyd_finite">this result</a>) – contradiction.


structure_id: BG_f
property_id: terminal object
type: category
proof:
Assume for contradiction that it has a terminal object. Then it would have a multi-terminal object (by <a href="/category-implication/multi-terminal_with_connected">this result</a>) – contradiction.


structure_id: BG_f
property_id: binary products
type: category
proof:
Assume for contradiction that it has binary products. Then it would have binary products and would have pullbacks, so it would have equalizers (by <a href="/category-implication/equalizers_criterion">this result</a>) – contradiction.


structure_id: BG_f
property_id: initial object
type: category
proof:
Assume for contradiction that it has an initial object. Then it would have a multi-initial object (by <a href="/category-implication/dual_multi-terminal_with_connected">this result</a>) – contradiction.


structure_id: BG_f
property_id: direct
type: category
proof:
Assume for contradiction that it is direct. Then it would be one-way (by <a href="/category-implication/direct_implies_one-way">this result</a>) – contradiction.


structure_id: BG_f
property_id: inverse
type: category
proof:
Assume for contradiction that it is inverse. Then it would be one-way (by <a href="/category-implication/dual_direct_implies_one-way">this result</a>) – contradiction.


structure_id: BG_c
property_id: essentially countable
type: category
proof:
Since it is countable, it is essentially countable (by <a href="/category-implication/countable_consequence">this result</a>).


structure_id: BG_c
property_id: finite copowers
type: category
proof:
Assume for contradiction that it has finite copowers. Then it would have filtered colimits and would have finite copowers, so it would have copowers (by <a href="/category-implication/copowers_criterion">this result</a>) – contradiction.


structure_id: BN
property_id: core-thin
type: category
proof:
Since it is gaunt, it is core-thin (by <a href="/category-implication/gaunt_characterization">this result</a>).


structure_id: BN
property_id: skeletal
type: category
proof:
Since it is gaunt, it is skeletal (by <a href="/category-implication/gaunt_characterization">this result</a>).


structure_id: BN
property_id: right cancellative
type: category
proof:
Since it is left cancellative and is self-dual, it is right cancellative (by <a href="/category-implication/self-dual_left cancellative">this result</a>).


structure_id: BN
property_id: locally cocartesian coclosed
type: category
proof:
Since it is locally cartesian closed and is self-dual, it is locally cocartesian coclosed (by <a href="/category-implication/self-dual_locally cartesian closed">this result</a>).


structure_id: BN
property_id: locally essentially small
type: category
proof:
Since it is accessible, it is locally essentially small (by <a href="/category-implication/accessible_locally_small">this result</a>).


structure_id: BN
property_id: pushouts
type: category
proof:
Since it is locally cocartesian coclosed, it has pushouts (by <a href="/category-implication/dual_pullbacks_are_local_products">this result</a>).


structure_id: BN
property_id: sequential colimits
type: category
proof:
Assume for contradiction that it has sequential colimits. Then it would be self-dual and would have sequential colimits, so it would have sequential limits (by <a href="/category-implication/self-dual_sequential colimits">this result</a>) – contradiction.


structure_id: BN
property_id: discrete
type: category
proof:
Assume for contradiction that it is discrete. Then it would be direct (by <a href="/category-implication/discrete_consequences">this result</a>) – contradiction.


structure_id: BN
property_id: balanced
type: category
proof:
Assume for contradiction that it is balanced. Then it would be balanced and would be left cancellative and would be right cancellative, so it would be a groupoid (by <a href="/category-implication/groupoid_criterion">this result</a>) – contradiction.


structure_id: BN
property_id: initial object
type: category
proof:
Assume for contradiction that it has an initial object. Then it would have an initial object and would have pushouts, so it would have binary coproducts (by <a href="/category-implication/dual_binary_products_criterion">this result</a>) – contradiction.


structure_id: BOn
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have pullbacks and would be self-dual, so it would have pushouts (by <a href="/category-implication/self-dual_pullbacks">this result</a>) – contradiction.


structure_id: 2
property_id: direct
type: category
proof:
Since it is discrete, it is direct (by <a href="/category-implication/discrete_consequences">this result</a>).


structure_id: 2
property_id: essentially discrete
type: category
proof:
Since it is discrete, it is essentially discrete (by <a href="/category-implication/discrete_consequences">this result</a>).


structure_id: 2
property_id: locally small
type: category
proof:
Since it is discrete, it is locally small (by <a href="/category-implication/discrete_consequences">this result</a>).


structure_id: 2
property_id: skeletal
type: category
proof:
Since it is discrete, it is skeletal (by <a href="/category-implication/discrete_consequences">this result</a>).


structure_id: 2
property_id: inverse
type: category
proof:
Since it is discrete, it is inverse (by <a href="/category-implication/dual_discrete_consequences">this result</a>).


structure_id: 2
property_id: groupoid
type: category
proof:
Since it is essentially discrete, it is a groupoid (by <a href="/category-implication/essentially_discrete_characterization">this result</a>).


structure_id: 2
property_id: thin
type: category
proof:
Since it is essentially discrete, it is thin (by <a href="/category-implication/essentially_discrete_characterization">this result</a>).


structure_id: 2
property_id: connected limits
type: category
proof:
Since it is essentially discrete, it has connected limits (by <a href="/category-implication/essentially_discrete_consequence">this result</a>).


structure_id: 2
property_id: locally essentially small
type: category
proof:
Since it is essentially discrete, it is locally essentially small (by <a href="/category-implication/essentially_discrete_consequence">this result</a>).


structure_id: 2
property_id: sequential limits
type: category
proof:
Since it is direct, it has sequential limits (by <a href="/category-implication/direct_implies_sequential_limits">this result</a>).


structure_id: 2
property_id: one-way
type: category
proof:
Since it is direct, it is one-way (by <a href="/category-implication/direct_implies_one-way">this result</a>).


structure_id: 2
property_id: connected colimits
type: category
proof:
Since it is essentially discrete, it has connected colimits (by <a href="/category-implication/dual_essentially_discrete_consequence">this result</a>).


structure_id: 2
property_id: sequential colimits
type: category
proof:
Since it is inverse, it has sequential colimits (by <a href="/category-implication/dual_direct_implies_sequential_limits">this result</a>).


structure_id: 2
property_id: reflexive coequalizers
type: category
proof:
Since it is one-way, it has reflexive coequalizers (by <a href="/category-implication/one-way_reflexive">this result</a>).


structure_id: 2
property_id: Cauchy complete
type: category
proof:
Since it has sequential colimits, it is Cauchy complete (by <a href="/category-implication/sequential_colimits_consequence">this result</a>).


structure_id: 2
property_id: regular-subobject-trivial
type: category
proof:
Since it is thin, it is regular-subobject-trivial (by <a href="/category-implication/thin_regular-subobject-trivial">this result</a>).


structure_id: 2
property_id: generator
type: category
proof:
Since it is inhabited and is thin, it has a generator (by <a href="/category-implication/thin_inhabited_consequence">this result</a>).


structure_id: 2
property_id: core-thin
type: category
proof:
Since it is one-way, it is core-thin (by <a href="/category-implication/one-way_implies_core-thin">this result</a>).


structure_id: 2
property_id: binary powers
type: category
proof:
Since it is thin, it has binary powers (by <a href="/category-implication/thin_binary_power">this result</a>).


structure_id: 2
property_id: coreflexive equalizers
type: category
proof:
Since it is one-way, it has coreflexive equalizers (by <a href="/category-implication/dual_one-way_reflexive">this result</a>).


structure_id: 2
property_id: coequalizers
type: category
proof:
Since it has connected colimits, it has coequalizers (by <a href="/category-implication/dual_connected_limits_characterization">this result</a>).


structure_id: 2
property_id: regular-quotient-trivial
type: category
proof:
Since it is thin, it is regular-quotient-trivial (by <a href="/category-implication/dual_thin_regular-subobject-trivial">this result</a>).


structure_id: 2
property_id: cogenerator
type: category
proof:
Since it is inhabited and is thin, it has a cogenerator (by <a href="/category-implication/dual_thin_inhabited_consequence">this result</a>).


structure_id: 2
property_id: binary copowers
type: category
proof:
Since it is thin, it has binary copowers (by <a href="/category-implication/dual_thin_binary_power">this result</a>).


structure_id: 2
property_id: filtered-colimit-stable monomorphisms
type: category
proof:
Since it is Cauchy complete and is essentially finite, it has filtered-colimit-stable monomorphisms (by <a href="/category-implication/finite_filtered_colimits">this result</a>).


structure_id: 2
property_id: gaunt
type: category
proof:
Since it is core-thin and is skeletal, it is gaunt (by <a href="/category-implication/gaunt_characterization">this result</a>).


structure_id: 2
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Since it is Cauchy complete and is essentially finite, it has cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_finite_filtered_colimits">this result</a>).


structure_id: 2
property_id: multi-cocomplete
type: category
proof:
Since it is locally finitely multi-presentable, it is multi-cocomplete (by <a href="/category-implication/locally_finitely_multi-presentable_another_definition">this result</a>).


structure_id: 2
property_id: multi-algebraic
type: category
proof:
Since it is a generalized variety and is multi-cocomplete, it is multi-algebraic (by <a href="/category-implication/multi-algebraic_another_definition">this result</a>).


structure_id: 2
property_id: multi-complete
type: category
proof:
Since it is multi-cocomplete and is self-dual, it is multi-complete (by <a href="/category-implication/self-dual_multi-cocomplete">this result</a>).


structure_id: 2
property_id: ℵ₁-filtered
type: category
proof:
Assume for contradiction that it is ℵ₁-filtered. Then it would be essentially countable and would be thin and would be ℵ₁-filtered, so it would have a terminal object (by <a href="/category-implication/aleph1-filtered-countable-thin">this result</a>) – contradiction.


structure_id: 2
property_id: ℵ₁-cofiltered
type: category
proof:
Assume for contradiction that it is ℵ₁-cofiltered. Then it would be essentially countable and would be thin and would be ℵ₁-cofiltered, so it would have an initial object (by <a href="/category-implication/dual_aleph1-filtered-countable-thin">this result</a>) – contradiction.


structure_id: 2
property_id: countable coproducts
type: category
proof:
Assume for contradiction that it has countable coproducts. Then it would have coequalizers and would have countable coproducts, so it would be ℵ₁-filtered (by <a href="/category-implication/aleph1-filtered_criterion">this result</a>) – contradiction.


structure_id: 2
property_id: countable products
type: category
proof:
Assume for contradiction that it has countable products. Then it would have finite products (by <a href="/category-implication/countable_products_consequence">this result</a>) – contradiction.


structure_id: 0
property_id: normal
type: category
proof:
Since it is mono-regular and is preadditive, it is normal (by <a href="/category-implication/normal_criterion">this result</a>).


structure_id: 0
property_id: conormal
type: category
proof:
Since it is epi-regular and is preadditive, it is conormal (by <a href="/category-implication/dual_normal_criterion">this result</a>).


structure_id: 0
property_id: binary coproducts
type: category
proof:
Since it has binary products and is self-dual, it has binary coproducts (by <a href="/category-implication/self-dual_binary products">this result</a>).


structure_id: 0
property_id: connected
type: category
proof:
Assume for contradiction that it is connected. Then it would be inhabited (by <a href="/category-implication/connected_consequence">this result</a>) – contradiction.


structure_id: 0
property_id: generator
type: category
proof:
Assume for contradiction that it has a generator. Then it would be inhabited (by <a href="/category-implication/generator_consequence">this result</a>) – contradiction.


structure_id: 0
property_id: cogenerator
type: category
proof:
Assume for contradiction that it has a cogenerator. Then it would be inhabited (by <a href="/category-implication/dual_generator_consequence">this result</a>) – contradiction.


structure_id: real_interval
property_id: locally copresentable
type: category
proof:
Since it is locally presentable and is self-dual, it is locally copresentable (by <a href="/category-implication/self-dual_locally presentable">this result</a>).


structure_id: real_interval
property_id: thin
type: category
proof:
Since it has copowers and is essentially small, it is thin (by <a href="/category-implication/dual_freyd_small">this result</a>).


structure_id: real_interval
property_id: Malcev
type: category
proof:
Since it is finitely complete and is thin, it is Malcev (by <a href="/category-implication/malcev_thin_condition">this result</a>).


structure_id: real_interval
property_id: natural numbers object
type: category
proof:
Since it has finite products and is thin, it has a natural numbers object (by <a href="/category-implication/nno_thin">this result</a>).


structure_id: real_interval
property_id: locally cartesian closed
type: category
proof:
Since it is semi-strongly connected and is thin, it is locally cartesian closed (by <a href="/category-implication/sequential_implies_lcc">this result</a>).


structure_id: real_interval
property_id: left cancellative
type: category
proof:
Since it is thin, it is left cancellative (by <a href="/category-implication/thin_consequences">this result</a>).


structure_id: real_interval
property_id: locally finite
type: category
proof:
Since it is thin, it is locally finite (by <a href="/category-implication/thin_consequences">this result</a>).


structure_id: real_interval
property_id: one-way
type: category
proof:
Since it is thin, it is one-way (by <a href="/category-implication/thin_consequences">this result</a>).


structure_id: real_interval
property_id: regular
type: category
proof:
Since it is finitely complete and is thin, it is regular (by <a href="/category-implication/thin_implies_regular">this result</a>).


structure_id: real_interval
property_id: co-Malcev
type: category
proof:
Since it is finitely cocomplete and is thin, it is co-Malcev (by <a href="/category-implication/dual_malcev_thin_condition">this result</a>).


structure_id: real_interval
property_id: locally cocartesian coclosed
type: category
proof:
Since it is semi-strongly connected and is thin, it is locally cocartesian coclosed (by <a href="/category-implication/dual_sequential_implies_lcc">this result</a>).


structure_id: real_interval
property_id: right cancellative
type: category
proof:
Since it is thin, it is right cancellative (by <a href="/category-implication/dual_thin_consequences">this result</a>).


structure_id: real_interval
property_id: coregular
type: category
proof:
Since it is finitely cocomplete and is thin, it is coregular (by <a href="/category-implication/dual_thin_implies_regular">this result</a>).


structure_id: real_interval
property_id: regular subobject classifier
type: category
proof:
Since it is finitely complete and is right cancellative, it has a regular subobject classifier (by <a href="/category-implication/regular_subobjects_trivial">this result</a>).


structure_id: real_interval
property_id: cocartesian coclosed
type: category
proof:
Since it has an initial object and is locally cocartesian coclosed, it is cocartesian coclosed (by <a href="/category-implication/dual_locally_cartesian_closed_with_terminal_is_closed">this result</a>).


structure_id: real_interval
property_id: strict terminal object
type: category
proof:
Since it is right cancellative and has a terminal object, it has a strict terminal object (by <a href="/category-implication/dual_strict_initial_left_criterion">this result</a>).


structure_id: real_interval
property_id: regular quotient object classifier
type: category
proof:
Since it is finitely cocomplete and is left cancellative, it has a regular quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_trivial">this result</a>).


structure_id: real_interval
property_id: countably distributive
type: category
proof:
Since it is cartesian closed and has countable coproducts, it is countably distributive (by <a href="/category-implication/countably_distributive_criterion">this result</a>).


structure_id: real_interval
property_id: cocartesian cofiltered limits
type: category
proof:
Since it is cocartesian coclosed and has cofiltered limits, it has cocartesian cofiltered limits (by <a href="/category-implication/dual_ccc_cartesian_filtered_colimits">this result</a>).


structure_id: real_interval
property_id: codistributive
type: category
proof:
Since it is cocartesian coclosed and has finite products, it is codistributive (by <a href="/category-implication/dual_distributive_criterion">this result</a>).


structure_id: real_interval
property_id: countably codistributive
type: category
proof:
Since it is cocartesian coclosed and has countable products, it is countably codistributive (by <a href="/category-implication/dual_countably_distributive_criterion">this result</a>).


structure_id: real_interval
property_id: infinitary codistributive
type: category
proof:
Since it is cocartesian coclosed and has products, it is infinitary codistributive (by <a href="/category-implication/dual_infinitary_distributive_criterion">this result</a>).


structure_id: real_interval
property_id: exact filtered colimits
type: category
proof:
Since it has cartesian filtered colimits and is thin, it has exact filtered colimits (by <a href="/category-implication/thin_exact_filtered_colimits">this result</a>).


structure_id: real_interval
property_id: exact cofiltered limits
type: category
proof:
Since it has cocartesian cofiltered limits and is thin, it has exact cofiltered limits (by <a href="/category-implication/dual_thin_exact_filtered_colimits">this result</a>).


structure_id: real_interval
property_id: finite
type: category
proof:
Assume for contradiction that it is finite. Then it would be finite and would be one-way and would be skeletal, so it would be direct (by <a href="/category-implication/direct_criterion">this result</a>) – contradiction.


structure_id: real_interval
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be essentially small and would be a groupoid, so it would be finitely accessible (by <a href="/category-implication/groupoids_are_generalized_varieties">this result</a>) – contradiction.


structure_id: real_interval
property_id: strongly connected
type: category
proof:
Assume for contradiction that it is strongly connected. Then it would be strongly connected and would be thin, so it would be trivial (by <a href="/category-implication/thin_not_strongly_connected">this result</a>) – contradiction.


structure_id: real_interval
property_id: disjoint finite coproducts
type: category
proof:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would be thin, so it would be trivial (by <a href="/category-implication/disjoint_coproducts_thin_trivial">this result</a>) – contradiction.


structure_id: real_interval
property_id: zero morphisms
type: category
proof:
Assume for contradiction that it has zero morphisms. Then it would be inhabited and would be thin and would have zero morphisms, so it would be trivial (by <a href="/category-implication/thin_zero_trivial">this result</a>) – contradiction.


structure_id: real_interval
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would have cofiltered-limit-stable epimorphisms and would have countable coproducts and would be an elementary topos, so it would be trivial (by <a href="/category-implication/topos_no_stable_epis">this result</a>) – contradiction.


structure_id: real_interval
property_id: disjoint finite products
type: category
proof:
Assume for contradiction that it has disjoint finite products. Then it would have disjoint finite products and would be thin, so it would be trivial (by <a href="/category-implication/dual_disjoint_coproducts_thin_trivial">this result</a>) – contradiction.


structure_id: N_oo
property_id: skeletal
type: category
proof:
Since it is direct, it is skeletal (by <a href="/category-implication/direct_implies_one-way">this result</a>).


structure_id: N_oo
property_id: Cauchy complete
type: category
proof:
Since it has sequential limits, it is Cauchy complete (by <a href="/category-implication/dual_sequential_colimits_consequence">this result</a>).


structure_id: N_oo
property_id: complete
type: category
proof:
Since it is cocomplete and is essentially small and is thin, it is complete (by <a href="/category-implication/dual_cocomplete_thin_criterion">this result</a>).


structure_id: N_oo
property_id: terminal object
type: category
proof:
Since it is essentially countable and is thin and is ℵ₁-filtered, it has a terminal object (by <a href="/category-implication/aleph1-filtered-countable-thin">this result</a>).


structure_id: N_oo
property_id: locally copresentable
type: category
proof:
Since it is coaccessible and is complete, it is locally copresentable (by <a href="/category-implication/dual_locally_presentable_definition">this result</a>).


structure_id: N_oo
property_id: discrete
type: category
proof:
Assume for contradiction that it is discrete. Then it would be inverse (by <a href="/category-implication/dual_discrete_consequences">this result</a>) – contradiction.


structure_id: N
property_id: generating set
type: category
proof:
Since it is thin, it has a generating set (by <a href="/category-implication/thin_consequences">this result</a>).


structure_id: N
property_id: cogenerating set
type: category
proof:
Since it is thin, it has a cogenerating set (by <a href="/category-implication/dual_thin_consequences">this result</a>).


structure_id: N
property_id: copowers
type: category
proof:
Since it has an initial object and is thin, it has copowers (by <a href="/category-implication/dual_thin_power">this result</a>).


structure_id: N
property_id: ℵ₂-small copowers
type: category
proof:
Since it has copowers, it has ℵ₂-small copowers (by <a href="/category-implication/dual_powers_include_aleph2-small_powers">this result</a>).


structure_id: N
property_id: countable copowers
type: category
proof:
Since it has ℵ₂-small copowers, it has countable copowers (by <a href="/category-implication/dual_aleph2-small-powers_include_countable_powers">this result</a>).


structure_id: N
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be self-dual and would have sequential limits, so it would have sequential colimits (by <a href="/category-implication/self-dual_sequential limits">this result</a>) – contradiction.


structure_id: N
property_id: finite
type: category
proof:
Assume for contradiction that it is finite. Then it would be finite and would be one-way and would be skeletal, so it would be inverse (by <a href="/category-implication/dual_direct_criterion">this result</a>) – contradiction.


structure_id: N
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be essentially finite and would have finite coproducts and would be thin, so it would have coproducts (by <a href="/category-implication/dual_thin_finite_product_reduction">this result</a>) – contradiction.


structure_id: N
property_id: complete
type: category
proof:
Assume for contradiction that it is complete. Then it would be complete and would be essentially small and would be thin, so it would be cocomplete (by <a href="/category-implication/cocomplete_thin_criterion">this result</a>) – contradiction.


structure_id: N
property_id: terminal object
type: category
proof:
Assume for contradiction that it has a terminal object. Then it would be multi-cocomplete and would have a terminal object, so it would be cocomplete (by <a href="/category-implication/dual_multi-complete_with_initial_object">this result</a>) – contradiction.


structure_id: N
property_id: additive
type: category
proof:
Assume for contradiction that it is additive. Then it would be additive and would have a regular quotient object classifier, so it would be trivial (by <a href="/category-implication/dual_additive_trivial_condition">this result</a>) – contradiction.


structure_id: On
property_id: coregular
type: category
proof:
Since it has equalizers and is finitely cocomplete and is locally cocartesian coclosed, it is coregular (by <a href="/category-implication/dual_lcc_implies_regular">this result</a>).


structure_id: On
property_id: accessible
type: category
proof:
Assume for contradiction that it is accessible. Then it would be accessible and would have pushouts, so it would be well-copowered (by <a href="/category-implication/accessible_well-copowered">this result</a>) – contradiction.


structure_id: On
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be essentially finite and would be filtered and would be thin, so it would have a terminal object (by <a href="/category-implication/filtered-finite-thin">this result</a>) – contradiction.


structure_id: On
property_id: essentially countable
type: category
proof:
Assume for contradiction that it is essentially countable. Then it would be essentially countable and would be thin and would be ℵ₁-filtered, so it would have a terminal object (by <a href="/category-implication/aleph1-filtered-countable-thin">this result</a>) – contradiction.


structure_id: On
property_id: essentially small
type: category
proof:
Assume for contradiction that it is essentially small. Then it would be well-copowered (by <a href="/category-implication/essentially_small_consequences">this result</a>) – contradiction.


structure_id: On
property_id: elementary topos
type: category
proof:
Assume for contradiction that it is an elementary topos. Then it would be an elementary topos and would be locally essentially small, so it would be well-copowered (by <a href="/category-implication/topos_well-copowered_criterion">this result</a>) – contradiction.


structure_id: On
property_id: coaccessible
type: category
proof:
Assume for contradiction that it is coaccessible. Then it would be well-copowered (by <a href="/category-implication/dual_accessible_well-powered">this result</a>) – contradiction.


structure_id: On
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be well-copowered (by <a href="/category-implication/dual_groupoid_consequence">this result</a>) – contradiction.


structure_id: On
property_id: pointed
type: category
proof:
Assume for contradiction that it is pointed. Then it would have a terminal object (by <a href="/category-implication/dual_pointed_characterization">this result</a>) – contradiction.


structure_id: On
property_id: quotient object classifier
type: category
proof:
Assume for contradiction that it has a quotient object classifier. Then it would be locally essentially small and would have a quotient object classifier, so it would be well-copowered (by <a href="/category-implication/dual_subobject_classifier_well-powered">this result</a>) – contradiction.


structure_id: On
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would have an initial object and would be self-dual, so it would have a terminal object (by <a href="/category-implication/self-dual_initial object">this result</a>) – contradiction.


structure_id: On
property_id: preadditive
type: category
proof:
Assume for contradiction that it is preadditive. Then it would have finite coproducts and would be preadditive, so it would have finite products (by <a href="/category-implication/preadditive_products_criterion">this result</a>) – contradiction.


structure_id: On
property_id: biproducts
type: category
proof:
Assume for contradiction that it has biproducts. Then it would have finite products (by <a href="/category-implication/biproducts_definition_consequences">this result</a>) – contradiction.


structure_id: On
property_id: epi-regular
type: category
proof:
Assume for contradiction that it is epi-regular. Then it would be epi-regular and would have a regular quotient object classifier, so it would have a quotient object classifier (by <a href="/category-implication/dual_regular_subobjects_suffice">this result</a>) – contradiction.


structure_id: Z_div
property_id: finite coproducts
type: category
proof:
Since it is distributive, it has finite coproducts (by <a href="/category-implication/distributive_assumption">this result</a>).


structure_id: Z_div
property_id: finite products
type: category
proof:
Since it is distributive, it has finite products (by <a href="/category-implication/distributive_assumption">this result</a>).


structure_id: Z_div
property_id: strict initial object
type: category
proof:
Since it is distributive, it has a strict initial object (by <a href="/category-implication/distributive_consequence">this result</a>).


structure_id: Z_div
property_id: codistributive
type: category
proof:
Since it is distributive and is thin, it is codistributive (by <a href="/category-implication/distributive_duality">this result</a>).


structure_id: Z_div
property_id: equalizers
type: category
proof:
Since it is thin, it has equalizers (by <a href="/category-implication/thin_regular-subobject-trivial">this result</a>).


structure_id: Z_div
property_id: coequalizers
type: category
proof:
Since it is thin, it has coequalizers (by <a href="/category-implication/dual_thin_regular-subobject-trivial">this result</a>).


structure_id: Z_div
property_id: strict terminal object
type: category
proof:
Since it is codistributive, it has a strict terminal object (by <a href="/category-implication/dual_distributive_consequence">this result</a>).


structure_id: Z_div
property_id: powers
type: category
proof:
Since it has a terminal object and is thin, it has powers (by <a href="/category-implication/thin_power">this result</a>).


structure_id: Z_div
property_id: countable powers
type: category
proof:
Since it has ℵ₂-small powers, it has countable powers (by <a href="/category-implication/aleph2-small-powers_include_countable_powers">this result</a>).


structure_id: Z_div
property_id: exact cofiltered limits
type: category
proof:
Assume for contradiction that it has exact cofiltered limits. Then it would have cocartesian cofiltered limits (by <a href="/category-implication/dual_exact_includes_cartesian_filtered_colimits">this result</a>) – contradiction.


structure_id: Z_div
property_id: semi-strongly connected
type: category
proof:
Assume for contradiction that it is semi-strongly connected. Then it would be semi-strongly connected and would be thin, so it would be locally cartesian closed (by <a href="/category-implication/sequential_implies_lcc">this result</a>) – contradiction.


structure_id: Z_div
property_id: essentially finite
type: category
proof:
Assume for contradiction that it is essentially finite. Then it would be Cauchy complete and would be essentially finite, so it would be finitely accessible (by <a href="/category-implication/finite_accessible">this result</a>) – contradiction.


structure_id: Delta
property_id: locally essentially small
type: category
proof:
Since it is locally finite, it is locally essentially small (by <a href="/category-implication/locally_finite_consequence">this result</a>).


structure_id: Delta
property_id: left cancellative
type: category
proof:
Assume for contradiction that it is left cancellative. Then it would have coreflexive equalizers (by <a href="/category-implication/reflexive_pair_trivial">this result</a>) – contradiction.


structure_id: Delta
property_id: equalizers
type: category
proof:
Assume for contradiction that it has equalizers. Then it would have coreflexive equalizers (by <a href="/category-implication/coreflexive_equalizers_are_equalizers">this result</a>) – contradiction.


structure_id: Delta
property_id: right cancellative
type: category
proof:
Assume for contradiction that it is right cancellative. Then it would have coreflexive equalizers (by <a href="/category-implication/dual_reflexive_pair_trivial">this result</a>) – contradiction.


structure_id: Delta
property_id: one-way
type: category
proof:
Assume for contradiction that it is one-way. Then it would have coreflexive equalizers (by <a href="/category-implication/dual_one-way_reflexive">this result</a>) – contradiction.


structure_id: Delta
property_id: cosifted limits
type: category
proof:
Assume for contradiction that it has cosifted limits. Then it would have coreflexive equalizers (by <a href="/category-implication/dual_sifted_colimits_consequence">this result</a>) – contradiction.


structure_id: Delta
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would have pushouts (by <a href="/category-implication/dual_groupoid_consequence">this result</a>) – contradiction.


structure_id: Delta
property_id: binary coproducts
type: category
proof:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would have coequalizers, so it would have pushouts (by <a href="/category-implication/dual_pullbacks_criterion">this result</a>) – contradiction.


structure_id: Delta
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be filtered and would be self-dual, so it would be cofiltered (by <a href="/category-implication/self-dual_filtered">this result</a>) – contradiction.


structure_id: Delta
property_id: cocomplete
type: category
proof:
Assume for contradiction that it is cocomplete. Then it would be accessible and would be cocomplete, so it would be locally presentable (by <a href="/category-implication/locally_presentable_definition">this result</a>) – contradiction.


structure_id: 1
property_id: Grothendieck topos
type: category
proof:
Since it is trivial, it is a Grothendieck topos (by <a href="/category-implication/trivial_consequence">this result</a>).


structure_id: 1
property_id: essentially discrete
type: category
proof:
Since it is trivial, it is essentially discrete (by <a href="/category-implication/trivial_consequence">this result</a>).


structure_id: 1
property_id: essentially finite
type: category
proof:
Since it is trivial, it is essentially finite (by <a href="/category-implication/trivial_consequence">this result</a>).


structure_id: 1
property_id: finitary algebraic
type: category
proof:
Since it is trivial, it is finitary algebraic (by <a href="/category-implication/trivial_consequence">this result</a>).


structure_id: 1
property_id: self-dual
type: category
proof:
Since it is trivial, it is self-dual (by <a href="/category-implication/trivial_consequence">this result</a>).


structure_id: 1
property_id: split abelian
type: category
proof:
Since it is trivial, it is split abelian (by <a href="/category-implication/trivial_consequence">this result</a>).


structure_id: 1
property_id: products
type: category
proof:
Since it has coproducts and is self-dual, it has products (by <a href="/category-implication/self-dual_coproducts">this result</a>).


structure_id: 1
property_id: exact cofiltered limits
type: category
proof:
Since it has exact filtered colimits and is self-dual, it has exact cofiltered limits (by <a href="/category-implication/self-dual_exact filtered colimits">this result</a>).


structure_id: 1
property_id: infinitary coextensive
type: category
proof:
Since it is infinitary extensive and is self-dual, it is infinitary coextensive (by <a href="/category-implication/self-dual_infinitary extensive">this result</a>).


structure_id: 1
property_id: cofiltered-limit-stable epimorphisms
type: category
proof:
Since it has exact cofiltered limits, it has cofiltered-limit-stable epimorphisms (by <a href="/category-implication/dual_exact_filtered_colimits_monos">this result</a>).


structure_id: 1
property_id: cocartesian cofiltered limits
type: category
proof:
Since it has exact cofiltered limits, it has cocartesian cofiltered limits (by <a href="/category-implication/dual_exact_includes_cartesian_filtered_colimits">this result</a>).


structure_id: 1
property_id: coextensive
type: category
proof:
Since it is infinitary coextensive, it is coextensive (by <a href="/category-implication/dual_infinitary_extensive_finitary">this result</a>).


structure_id: 1
property_id: cocartesian coclosed
type: category
proof:
Since it is cartesian closed and is self-dual, it is cocartesian coclosed (by <a href="/category-implication/self-dual_cartesian closed">this result</a>).


structure_id: 1
property_id: disjoint finite products
type: category
proof:
Since it has disjoint finite coproducts and is self-dual, it has disjoint finite products (by <a href="/category-implication/self-dual_disjoint finite coproducts">this result</a>).


structure_id: 1
property_id: quotient object classifier
type: category
proof:
Since it is self-dual and has a subobject classifier, it has a quotient object classifier (by <a href="/category-implication/self-dual_subobject classifier">this result</a>).


structure_id: 1
property_id: terminal object
type: category
proof:
Since it is essentially finite and is filtered and is thin, it has a terminal object (by <a href="/category-implication/filtered-finite-thin">this result</a>).


structure_id: 1
property_id: CSP
type: category
proof:
Since it has biproducts and has cofiltered limits and has cofiltered-limit-stable epimorphisms and has coproducts, it satisfies CSP (by <a href="/category-implication/dual_CIP_criterion">this result</a>).


structure_id: 1
property_id: strict terminal object
type: category
proof:
Since it is coextensive, it has a strict terminal object (by <a href="/category-implication/dual_extensive_consequences">this result</a>).


structure_id: 1
property_id: countably codistributive
type: category
proof:
Since it is infinitary codistributive, it is countably codistributive (by <a href="/category-implication/dual_infinitary_distributive_trivial">this result</a>).


structure_id: walking_commutative_square
property_id: complete
type: category
proof:
Since it is cocomplete and is self-dual, it is complete (by <a href="/category-implication/self-dual_cocomplete">this result</a>).


structure_id: walking_commutative_square
property_id: coregular
type: category
proof:
Since it is regular and is self-dual, it is coregular (by <a href="/category-implication/self-dual_regular">this result</a>).


structure_id: walking_commutative_square
property_id: cosifted limits
type: category
proof:
Since it is self-dual and has sifted colimits, it has cosifted limits (by <a href="/category-implication/self-dual_sifted colimits">this result</a>).


structure_id: walking_commutative_square
property_id: strict initial object
type: category
proof:
Since it is cartesian closed and has an initial object, it has a strict initial object (by <a href="/category-implication/ccc_consequence">this result</a>).


structure_id: walking_commutative_square
property_id: direct
type: category
proof:
Since it is finite and is one-way and is skeletal, it is direct (by <a href="/category-implication/direct_criterion">this result</a>).


structure_id: walking_commutative_square
property_id: strict terminal object
type: category
proof:
Since it is cocartesian coclosed and has a terminal object, it has a strict terminal object (by <a href="/category-implication/dual_ccc_consequence">this result</a>).


structure_id: walking_commutative_square
property_id: inverse
type: category
proof:
Since it is finite and is one-way and is skeletal, it is inverse (by <a href="/category-implication/dual_direct_criterion">this result</a>).


structure_id: walking_commutative_square
property_id: quotient object classifier
type: category
proof:
Assume for contradiction that it has a quotient object classifier. Then it would have a quotient object classifier and would be self-dual, so it would have a subobject classifier (by <a href="/category-implication/self-dual_quotient object classifier">this result</a>) – contradiction.


structure_id: walking_composable_pair
property_id: cofiltered limits
type: category
proof:
Since it has exact cofiltered limits, it has cofiltered limits (by <a href="/category-implication/dual_exact_filtered_colimits_condition">this result</a>).


structure_id: walking_composable_pair
property_id: groupoid
type: category
proof:
Assume for contradiction that it is a groupoid. Then it would be a groupoid and would be thin, so it would be essentially discrete (by <a href="/category-implication/essentially_discrete_characterization">this result</a>) – contradiction.


structure_id: walking_coreflexive_pair
property_id: finitely accessible
type: category
proof:
Since it is Cauchy complete and is essentially finite, it is finitely accessible (by <a href="/category-implication/finite_accessible">this result</a>).


structure_id: walking_coreflexive_pair
property_id: filtered colimits
type: category
proof:
Since it is Cauchy complete and is essentially finite, it has filtered colimits (by <a href="/category-implication/finite_filtered_colimits">this result</a>).


structure_id: walking_coreflexive_pair
property_id: cofiltered limits
type: category
proof:
Since it is Cauchy complete and is essentially finite, it has cofiltered limits (by <a href="/category-implication/dual_finite_filtered_colimits">this result</a>).


structure_id: walking_coreflexive_pair
property_id: pullbacks
type: category
proof:
Assume for contradiction that it has pullbacks. Then it would have cofiltered limits and would have pullbacks, so it would have wide pullbacks (by <a href="/category-implication/wide_pullbacks_criterion">this result</a>) – contradiction.


structure_id: walking_coreflexive_pair
property_id: finite products
type: category
proof:
Assume for contradiction that it has finite products. Then it would have finite powers (by <a href="/category-implication/finite_products_include_finite_powers">this result</a>) – contradiction.


structure_id: walking_fork
property_id: generalized variety
type: category
proof:
Since it is finitely accessible and is left cancellative, it is a generalized variety (by <a href="/category-implication/generalized_variety_left_cancellative">this result</a>).


structure_id: walking_fork
property_id: disjoint finite coproducts
type: category
proof:
Assume for contradiction that it has disjoint finite coproducts. Then it would have disjoint finite coproducts and would be locally cartesian closed, so it would be extensive (by <a href="/category-implication/lcc_implies_extensive">this result</a>) – contradiction.


structure_id: walking_fork
property_id: finite coproducts
type: category
proof:
Assume for contradiction that it has finite coproducts. Then it would have finite copowers (by <a href="/category-implication/dual_finite_products_include_finite_powers">this result</a>) – contradiction.


structure_id: walking_fork
property_id: self-dual
type: category
proof:
Assume for contradiction that it is self-dual. Then it would be cosifted and would be self-dual, so it would be sifted (by <a href="/category-implication/self-dual_cosifted">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: coreflexive equalizers
type: category
proof:
Since it is subobject-trivial, it has coreflexive equalizers (by <a href="/category-implication/reflexive_pair_trivial_2">this result</a>).


structure_id: walking_idempotent
property_id: reflexive coequalizers
type: category
proof:
Since it is subobject-trivial, it has reflexive coequalizers (by <a href="/category-implication/reflexive_pair_trivial_2">this result</a>).


structure_id: walking_idempotent
property_id: ℵ₁-cofiltered
type: category
proof:
Since it is self-dual and is ℵ₁-filtered, it is ℵ₁-cofiltered (by <a href="/category-implication/self-dual_ℵ₁-filtered">this result</a>).


structure_id: walking_idempotent
property_id: quotient-trivial
type: category
proof:
Since it is self-dual and is subobject-trivial, it is quotient-trivial (by <a href="/category-implication/self-dual_subobject-trivial">this result</a>).


structure_id: walking_idempotent
property_id: connected
type: category
proof:
Since it is sifted, it is connected (by <a href="/category-implication/sifted_is_connected">this result</a>).


structure_id: walking_idempotent
property_id: thin
type: category
proof:
Assume for contradiction that it is thin. Then it would be essentially finite and would be filtered and would be thin, so it would have a terminal object (by <a href="/category-implication/filtered-finite-thin">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: initial object
type: category
proof:
Assume for contradiction that it has an initial object. Then it would have an initial object and would be self-dual, so it would have a terminal object (by <a href="/category-implication/self-dual_initial object">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: finite coproducts
type: category
proof:
Assume for contradiction that it has finite coproducts. Then it would have finite coproducts and would be preadditive, so it would have finite products (by <a href="/category-implication/preadditive_products_criterion">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: binary products
type: category
proof:
Assume for contradiction that it has binary products. Then it would have binary products and would have coreflexive equalizers, so it would have equalizers (by <a href="/category-implication/equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: kernels
type: category
proof:
Assume for contradiction that it has kernels. Then it would have kernels and would be preadditive, so it would have equalizers (by <a href="/category-implication/equalizers_via_kernels">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: binary coproducts
type: category
proof:
Assume for contradiction that it has binary coproducts. Then it would have binary coproducts and would have reflexive coequalizers, so it would have coequalizers (by <a href="/category-implication/dual_equalizers_via_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: Grothendieck topos
type: category
proof:
Assume for contradiction that it is a Grothendieck topos. Then it would be locally presentable (by <a href="/category-implication/grothendieck_topos_consequence">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: pullbacks
type: category
proof:
Assume for contradiction that it has pullbacks. Then it would be essentially finite and would have pullbacks, so it would have wide pullbacks (by <a href="/category-implication/wide_pullbacks_finite">this result</a>) – contradiction.


structure_id: walking_idempotent
property_id: pushouts
type: category
proof:
Assume for contradiction that it has pushouts. Then it would be essentially finite and would have pushouts, so it would have wide pushouts (by <a href="/category-implication/dual_wide_pullbacks_finite">this result</a>) – contradiction.


structure_id: walking_pair
property_id: sifted colimits
type: category
proof:
Since it has filtered colimits and is left cancellative, it has sifted colimits (by <a href="/category-implication/sifted_colimits_left_cancellative">this result</a>).


structure_id: walking_pair
property_id: cosifted limits
type: category
proof:
Since it has cofiltered limits and is right cancellative, it has cosifted limits (by <a href="/category-implication/dual_sifted_colimits_left_cancellative">this result</a>).


structure_id: walking_pair
property_id: pushouts
type: category
proof:
Assume for contradiction that it has pushouts. Then it would have pushouts and would be self-dual, so it would have pullbacks (by <a href="/category-implication/self-dual_pushouts">this result</a>) – contradiction.


structure_id: walking_pair
property_id: multi-complete
type: category
proof:
Assume for contradiction that it is multi-complete. Then it would be multi-complete and would be self-dual, so it would be multi-cocomplete (by <a href="/category-implication/self-dual_multi-complete">this result</a>) – contradiction.


structure_id: walking_span
property_id: finite copowers
type: category
proof:
Since it has countable copowers, it has finite copowers (by <a href="/category-implication/dual_countable_powers_include_finite_powers">this result</a>).


structure_id: walking_span
property_id: semi-strongly connected
type: category
proof:
Assume for contradiction that it is semi-strongly connected. Then it would have coequalizers and would be semi-strongly connected, so it would be filtered (by <a href="/category-implication/filtered_via_coequalizers">this result</a>) – contradiction.


structure_id: walking_splitting
property_id: coequalizers
type: category
proof:
Since it has equalizers and is self-dual, it has coequalizers (by <a href="/category-implication/self-dual_equalizers">this result</a>).


structure_id: walking_splitting
property_id: conormal
type: category
proof:
Since it is normal and is self-dual, it is conormal (by <a href="/category-implication/self-dual_normal">this result</a>).


structure_id: abelianization
property_id: left adjoint
type: functor
proof:
Since it is a reflector, it is a left adjoint (by <a href="/functor-implication/reflector_consequences">this result</a>).


structure_id: abelianization
property_id: right-invertible
type: functor
proof:
Since it is a reflector, it is right-invertible (by <a href="/functor-implication/reflector_consequences">this result</a>).


structure_id: abelianization
property_id: preserves binary products
type: functor
proof:
Since it preserves finite products, it preserves binary products (by <a href="/functor-implication/preserve_finite_products_consequences">this result</a>).


structure_id: abelianization
property_id: preserves terminal objects
type: functor
proof:
Since it preserves finite products, it preserves terminal objects (by <a href="/functor-implication/preserve_finite_products_consequences">this result</a>).


structure_id: abelianization
property_id: preserves initial objects
type: functor
proof:
Since it preserves terminal objects, and the source has the required properties (pointed), and the target has the required properties (pointed), it preserves initial objects (by <a href="/functor-implication/zero_preserving_condition">this result</a>).


structure_id: abelianization
property_id: essentially surjective
type: functor
proof:
Since it is right-invertible, it is essentially surjective (by <a href="/functor-implication/right-invertible_consequences">this result</a>).


structure_id: abelianization
property_id: cocontinuous
type: functor
proof:
Since it is a left adjoint, it is cocontinuous (by <a href="/functor-implication/dual_right_adjoint_consequences">this result</a>).


structure_id: abelianization
property_id: dominant
type: functor
proof:
Since it is essentially surjective, it is dominant (by <a href="/functor-implication/essentially-surjective_implies_dominant">this result</a>).


structure_id: abelianization
property_id: finitary
type: functor
proof:
Since it is cocontinuous, it is finitary (by <a href="/functor-implication/dual_continuous_consequences">this result</a>).


structure_id: abelianization
property_id: preserves coproducts
type: functor
proof:
Since it is cocontinuous, it preserves coproducts (by <a href="/functor-implication/dual_continuous_consequences">this result</a>).


structure_id: abelianization
property_id: right exact
type: functor
proof:
Since it is cocontinuous, it is right exact (by <a href="/functor-implication/dual_continuous_consequences">this result</a>).


structure_id: abelianization
property_id: preserves finite coproducts
type: functor
proof:
Since it preserves coproducts, it preserves finite coproducts (by <a href="/functor-implication/dual_products_consequences">this result</a>).


structure_id: abelianization
property_id: preserves coequalizers
type: functor
proof:
Since it is right exact, it preserves coequalizers (by <a href="/functor-implication/dual_left_exact_consequences">this result</a>).


structure_id: abelianization
property_id: preserves epimorphisms
type: functor
proof:
Since it is right exact, it preserves epimorphisms (by <a href="/functor-implication/dual_left_exact_consequences">this result</a>).


structure_id: abelianization
property_id: preserves binary coproducts
type: functor
proof:
Since it preserves finite coproducts, it preserves binary coproducts (by <a href="/functor-implication/dual_preserve_finite_products_consequences">this result</a>).


structure_id: abelianization
property_id: preserves reflexive coequalizers
type: functor
proof:
Since it preserves coequalizers, it preserves reflexive coequalizers (by <a href="/functor-implication/dual_equalizers_preservation_consequences">this result</a>).


structure_id: abelianization
property_id: preserves regular epimorphisms
type: functor
proof:
Since it preserves coequalizers, it preserves regular epimorphisms (by <a href="/functor-implication/dual_equalizers_preservation_consequences">this result</a>).


structure_id: abelianization
property_id: continuous
type: functor
proof:
Assume for contradiction that it is continuous. Then it would preserve products (by <a href="/functor-implication/continuous_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: cofinitary
type: functor
proof:
Assume for contradiction that it is cofinitary. Then it would be cofinitary and would preserve finite products, and the source has the required properties (finite products), so it would preserve products (by <a href="/functor-implication/product_criterion_filtered">this result</a>) – contradiction.


structure_id: abelianization
property_id: left exact
type: functor
proof:
Assume for contradiction that it is left exact. Then it would preserve monomorphisms (by <a href="/functor-implication/left_exact_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: preserves regular monomorphisms
type: functor
proof:
Assume for contradiction that it preserves regular monomorphisms. Then it would preserve regular monomorphisms, and the source has the required properties (mono-regular), so it would preserve monomorphisms (by <a href="/functor-implication/mono-preserving_criterion">this result</a>) – contradiction.


structure_id: abelianization
property_id: fully faithful
type: functor
proof:
Assume for contradiction that it is fully faithful. Then it would be faithful (by <a href="/functor-implication/fully-faithful-definition">this result</a>) – contradiction.


structure_id: abelianization
property_id: left-invertible
type: functor
proof:
Assume for contradiction that it is left-invertible. Then it would be essentially injective (by <a href="/functor-implication/left-invertible_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: full on isomorphisms
type: functor
proof:
Assume for contradiction that it is full on isomorphisms. Then it would be essentially injective (by <a href="/functor-implication/full-on-isomorphisms_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: pseudomonic
type: functor
proof:
Assume for contradiction that it is pseudomonic. Then it would be faithful (by <a href="/functor-implication/pseudomonic_definition">this result</a>) – contradiction.


structure_id: abelianization
property_id: monadic
type: functor
proof:
Assume for contradiction that it is monadic. Then it would be faithful (by <a href="/functor-implication/monadic_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: conservative
type: functor
proof:
Assume for contradiction that it is conservative. Then it would be conservative and would preserve coequalizers, and the source has the required properties (coequalizers), so it would be faithful (by <a href="/functor-implication/dual_faithful-via-equalizers">this result</a>) – contradiction.


structure_id: abelianization
property_id: comonadic
type: functor
proof:
Assume for contradiction that it is comonadic. Then it would be faithful (by <a href="/functor-implication/dual_monadic_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: right adjoint
type: functor
proof:
Assume for contradiction that it is a right adjoint. Then it would be continuous (by <a href="/functor-implication/right_adjoint_consequences">this result</a>) – contradiction.


structure_id: abelianization
property_id: equivalence
type: functor
proof:
Assume for contradiction that it is an equivalence. Then it would be fully faithful (by <a href="/functor-implication/equivalence_criterion">this result</a>) – contradiction.


structure_id: abelianization
property_id: exact
type: functor
proof:
Assume for contradiction that it is exact. Then it would be left exact (by <a href="/functor-implication/exact_definition">this result</a>) – contradiction.


structure_id: abelianization
property_id: preserves equalizers
type: functor
proof:
Assume for contradiction that it preserves equalizers. Then it would preserve equalizers and would preserve finite products, and the source has the required properties (finite products), so it would be left exact (by <a href="/functor-implication/left_exact_criterion">this result</a>) – contradiction.


structure_id: abelianization
property_id: preserves coreflexive equalizers
type: functor
proof:
Assume for contradiction that it preserves coreflexive equalizers. Then it would preserve coreflexive equalizers, and the source has the required properties (pushouts), so it would preserve regular monomorphisms (by <a href="/functor-implication/another_regular-mono-preserving_criterion">this result</a>) – contradiction.


structure_id: abelianization
property_id: regular
type: functor
proof:
Assume for contradiction that it is regular. Then it would be left exact (by <a href="/functor-implication/regular_functor_definition">this result</a>) – contradiction.


structure_id: abelianization
property_id: coregular
type: functor
proof:
Assume for contradiction that it is coregular. Then it would preserve regular monomorphisms (by <a href="/functor-implication/dual_regular_functor_definition">this result</a>) – contradiction.


structure_id: abelianization
property_id: isomorphism
type: functor
proof:
Assume for contradiction that it is an isomorphism. Then it would be an equivalence (by <a href="/functor-implication/isomorphism_is_equivalence">this result</a>) – contradiction.


structure_id: abelianization
property_id: coreflector
type: functor
proof:
Assume for contradiction that it is a coreflector. Then it would be a right adjoint (by <a href="/functor-implication/dual_reflector_consequences">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: preserves coreflexive equalizers
type: functor
proof:
Since it preserves equalizers, it preserves coreflexive equalizers (by <a href="/functor-implication/equalizers_preservation_consequences">this result</a>).


structure_id: binary_coproduct_sets
property_id: preserves regular monomorphisms
type: functor
proof:
Since it preserves equalizers, it preserves regular monomorphisms (by <a href="/functor-implication/equalizers_preservation_consequences">this result</a>).


structure_id: binary_coproduct_sets
property_id: faithful
type: functor
proof:
Since it is conservative and preserves equalizers, and the source has the required properties (equalizers), it is faithful (by <a href="/functor-implication/faithful-via-equalizers">this result</a>).


structure_id: binary_coproduct_sets
property_id: preserves monomorphisms
type: functor
proof:
Since it preserves regular monomorphisms, and the source has the required properties (mono-regular), it preserves monomorphisms (by <a href="/functor-implication/mono-preserving_criterion">this result</a>).


structure_id: binary_coproduct_sets
property_id: comonadic
type: functor
proof:
Since it is conservative and is a left adjoint and preserves coreflexive equalizers, and the source has the required properties (coreflexive equalizers), it is comonadic (by <a href="/functor-implication/dual_crude_monadicity_theorem">this result</a>).


structure_id: binary_coproduct_sets
property_id: coregular
type: functor
proof:
Since it preserves regular monomorphisms and is right exact, it is coregular (by <a href="/functor-implication/dual_regular_functor_definition">this result</a>).


structure_id: binary_coproduct_sets
property_id: preserves initial objects
type: functor
proof:
Since it preserves finite coproducts, it preserves initial objects (by <a href="/functor-implication/dual_preserve_finite_products_consequences">this result</a>).


structure_id: binary_coproduct_sets
property_id: equivalence
type: functor
proof:
Assume for contradiction that it is an equivalence. Then it would be a reflector (by <a href="/functor-implication/equivalence_consequences">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: preserves finite products
type: functor
proof:
Assume for contradiction that it preserves finite products. Then it would preserve binary products (by <a href="/functor-implication/preserve_finite_products_consequences">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: fully faithful
type: functor
proof:
Assume for contradiction that it is fully faithful. Then it would be essentially surjective and would be fully faithful, so it would be an equivalence (by <a href="/functor-implication/equivalence_criterion">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: preserves products
type: functor
proof:
Assume for contradiction that it preserves products. Then it would preserve finite products (by <a href="/functor-implication/products_consequences">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: left exact
type: functor
proof:
Assume for contradiction that it is left exact. Then it would preserve finite products (by <a href="/functor-implication/left_exact_consequences">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: full
type: functor
proof:
Assume for contradiction that it is full. Then it would be conservative and would be full, so it would be full on isomorphisms (by <a href="/functor-implication/full-on-isomorphisms_criterion">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: pseudomonic
type: functor
proof:
Assume for contradiction that it is pseudomonic. Then it would be full on isomorphisms (by <a href="/functor-implication/pseudomonic_definition">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: continuous
type: functor
proof:
Assume for contradiction that it is continuous. Then it would be left exact (by <a href="/functor-implication/continuous_consequences">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: representable
type: functor
proof:
Assume for contradiction that it is representable. Then it would be continuous (by <a href="/functor-implication/representable_is_continuous">this result</a>) – contradiction.


structure_id: binary_coproduct_sets
property_id: monadic
type: functor
proof:
Assume for contradiction that it is monadic. Then it would be a right adjoint (by <a href="/functor-implication/monadic_consequences">this result</a>) – contradiction.


structure_id: diagonal_sets
property_id: continuous
type: functor
proof:
Since it is a right adjoint, it is continuous (by <a href="/functor-implication/right_adjoint_consequences">this result</a>).


structure_id: diagonal_sets
property_id: conservative
type: functor
proof:
Since it is left-invertible, it is conservative (by <a href="/functor-implication/left-invertible_consequences">this result</a>).


structure_id: diagonal_sets
property_id: essentially injective
type: functor
proof:
Since it is left-invertible, it is essentially injective (by <a href="/functor-implication/left-invertible_consequences">this result</a>).


structure_id: diagonal_sets
property_id: faithful
type: functor
proof:
Since it is left-invertible, it is faithful (by <a href="/functor-implication/left-invertible_consequences">this result</a>).


structure_id: diagonal_sets
property_id: cofinitary
type: functor
proof:
Since it is continuous, it is cofinitary (by <a href="/functor-implication/continuous_consequences">this result</a>).


structure_id: diagonal_sets
property_id: left exact
type: functor
proof:
Since it is continuous, it is left exact (by <a href="/functor-implication/continuous_consequences">this result</a>).


structure_id: diagonal_sets
property_id: preserves products
type: functor
proof:
Since it is continuous, it preserves products (by <a href="/functor-implication/continuous_consequences">this result</a>).


structure_id: diagonal_sets
property_id: preserves finite products
type: functor
proof:
Since it preserves products, it preserves finite products (by <a href="/functor-implication/products_consequences">this result</a>).


structure_id: diagonal_sets
property_id: exact
type: functor
proof:
Since it is left exact and is right exact, it is exact (by <a href="/functor-implication/exact_definition">this result</a>).


structure_id: diagonal_sets
property_id: preserves equalizers
type: functor
proof:
Since it is left exact, it preserves equalizers (by <a href="/functor-implication/left_exact_consequences">this result</a>).


structure_id: diagonal_sets
property_id: preserves monomorphisms
type: functor
proof:
Since it is left exact, it preserves monomorphisms (by <a href="/functor-implication/left_exact_consequences">this result</a>).


structure_id: diagonal_sets
property_id: regular
type: functor
proof:
Since it is left exact and preserves regular epimorphisms, it is regular (by <a href="/functor-implication/regular_functor_definition">this result</a>).


structure_id: diagonal_sets
property_id: monadic
type: functor
proof:
Since it is conservative and preserves reflexive coequalizers and is a right adjoint, and the source has the required properties (reflexive coequalizers), it is monadic (by <a href="/functor-implication/crude_monadicity_theorem">this result</a>).


structure_id: diagonal_sets
property_id: essentially surjective
type: functor
proof:
Assume for contradiction that it is essentially surjective. Then it would be dominant (by <a href="/functor-implication/essentially-surjective_implies_dominant">this result</a>) – contradiction.


structure_id: diagonal_sets
property_id: equivalence
type: functor
proof:
Assume for contradiction that it is an equivalence. Then it would be essentially surjective (by <a href="/functor-implication/equivalence_criterion">this result</a>) – contradiction.


structure_id: diagonal_sets
property_id: fully faithful
type: functor
proof:
Assume for contradiction that it is fully faithful. Then it would be full (by <a href="/functor-implication/fully-faithful-definition">this result</a>) – contradiction.


structure_id: diagonal_sets
property_id: right-invertible
type: functor
proof:
Assume for contradiction that it is right-invertible. Then it would be essentially surjective (by <a href="/functor-implication/right-invertible_consequences">this result</a>) – contradiction.


structure_id: diagonal_sets
property_id: reflector
type: functor
proof:
Assume for contradiction that it is a reflector. Then it would be right-invertible (by <a href="/functor-implication/reflector_consequences">this result</a>) – contradiction.


structure_id: diagonal_sets
property_id: coreflector
type: functor
proof:
Assume for contradiction that it is a coreflector. Then it would be right-invertible (by <a href="/functor-implication/dual_reflector_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: continuous
type: functor
proof:
Since it is representable, it is continuous (by <a href="/functor-implication/representable_is_continuous">this result</a>).


structure_id: binary_product_sets
property_id: preserves regular epimorphisms
type: functor
proof:
Since it preserves reflexive coequalizers, and the source has the required properties (pullbacks), it preserves regular epimorphisms (by <a href="/functor-implication/dual_another_regular-mono-preserving_criterion">this result</a>).


structure_id: binary_product_sets
property_id: right adjoint
type: functor
proof:
Since it is continuous, and the source has the required properties (cogenerating set, complete, locally essentially small, well-powered), and the target has the required properties (locally essentially small), it is a right adjoint (by <a href="/functor-implication/saft">this result</a>).


structure_id: binary_product_sets
property_id: preserves epimorphisms
type: functor
proof:
Since it preserves regular epimorphisms, and the source has the required properties (epi-regular), it preserves epimorphisms (by <a href="/functor-implication/dual_mono-preserving_criterion">this result</a>).


structure_id: binary_product_sets
property_id: conservative
type: functor
proof:
Assume for contradiction that it is conservative. Then it would be conservative and would preserve equalizers, and the source has the required properties (equalizers), so it would be faithful (by <a href="/functor-implication/faithful-via-equalizers">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: equivalence
type: functor
proof:
Assume for contradiction that it is an equivalence. Then it would be a coreflector (by <a href="/functor-implication/dual_equivalence_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: right exact
type: functor
proof:
Assume for contradiction that it is right exact. Then it would preserve coequalizers (by <a href="/functor-implication/dual_left_exact_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: preserves binary coproducts
type: functor
proof:
Assume for contradiction that it preserves binary coproducts. Then it would preserve binary coproducts and would preserve reflexive coequalizers, and the source has the required properties (binary coproducts), so it would preserve coequalizers (by <a href="/functor-implication/dual_criterion_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: exact
type: functor
proof:
Assume for contradiction that it is exact. Then it would be right exact (by <a href="/functor-implication/exact_definition">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: cocontinuous
type: functor
proof:
Assume for contradiction that it is cocontinuous. Then it would be right exact (by <a href="/functor-implication/dual_continuous_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: preserves finite coproducts
type: functor
proof:
Assume for contradiction that it preserves finite coproducts. Then it would preserve binary coproducts (by <a href="/functor-implication/dual_preserve_finite_products_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: coregular
type: functor
proof:
Assume for contradiction that it is coregular. Then it would be right exact (by <a href="/functor-implication/dual_regular_functor_definition">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: left adjoint
type: functor
proof:
Assume for contradiction that it is a left adjoint. Then it would be cocontinuous (by <a href="/functor-implication/dual_right_adjoint_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: preserves coproducts
type: functor
proof:
Assume for contradiction that it preserves coproducts. Then it would preserve finite coproducts (by <a href="/functor-implication/dual_products_consequences">this result</a>) – contradiction.


structure_id: binary_product_sets
property_id: reflector
type: functor
proof:
Assume for contradiction that it is a reflector. Then it would be a left adjoint (by <a href="/functor-implication/reflector_consequences">this result</a>) – contradiction.


structure_id: power_set_contravariant
property_id: faithful
type: functor
proof:
Since it is monadic, it is faithful (by <a href="/functor-implication/monadic_consequences">this result</a>).


structure_id: power_set_contravariant
property_id: equivalence
type: functor
proof:
Assume for contradiction that it is an equivalence. Then it would be left-invertible (by <a href="/functor-implication/equivalence_invertible">this result</a>) – contradiction.


structure_id: power_set_contravariant
property_id: cocontinuous
type: functor
proof:
Assume for contradiction that it is cocontinuous. Then it would be finitary (by <a href="/functor-implication/dual_continuous_consequences">this result</a>) – contradiction.


structure_id: power_set_contravariant
property_id: preserves finite coproducts
type: functor
proof:
Assume for contradiction that it preserves finite coproducts. Then it would preserve initial objects (by <a href="/functor-implication/dual_preserve_finite_products_consequences">this result</a>) – contradiction.


structure_id: power_set_contravariant
property_id: comonadic
type: functor
proof:
Assume for contradiction that it is comonadic. Then it would be a left adjoint (by <a href="/functor-implication/dual_monadic_consequences">this result</a>) – contradiction.


structure_id: power_set_covariant
property_id: preserves regular monomorphisms
type: functor
proof:
Since it preserves monomorphisms, and the target has the required properties (mono-regular), it preserves regular monomorphisms (by <a href="/functor-implication/regular-mono-preserving_criterion">this result</a>).


structure_id: power_set_covariant
property_id: preserves regular epimorphisms
type: functor
proof:
Since it preserves epimorphisms, and the target has the required properties (epi-regular), it preserves regular epimorphisms (by <a href="/functor-implication/dual_regular-mono-preserving_criterion">this result</a>).


structure_id: power_set_covariant
property_id: continuous
type: functor
proof:
Assume for contradiction that it is continuous. Then it would be cofinitary (by <a href="/functor-implication/continuous_consequences">this result</a>) – contradiction.


structure_id: power_set_covariant
property_id: preserves finite products
type: functor
proof:
Assume for contradiction that it preserves finite products. Then it would preserve terminal objects (by <a href="/functor-implication/preserve_finite_products_consequences">this result</a>) – contradiction.


structure_id: power_set_covariant
property_id: left exact
type: functor
proof:
Assume for contradiction that it is left exact. Then it would preserve equalizers (by <a href="/functor-implication/left_exact_consequences">this result</a>) – contradiction.


structure_id: power_set_covariant
property_id: preserves binary products
type: functor
proof:
Assume for contradiction that it preserves binary products. Then it would preserve binary products and would preserve coreflexive equalizers, and the source has the required properties (binary products), so it would preserve equalizers (by <a href="/functor-implication/criterion_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: power_set_covariant
property_id: preserves coequalizers
type: functor
proof:
Assume for contradiction that it preserves coequalizers. Then it would preserve reflexive coequalizers (by <a href="/functor-implication/dual_equalizers_preservation_consequences">this result</a>) – contradiction.


structure_id: discrete_topology
property_id: left exact
type: functor
proof:
Since it preserves equalizers and preserves finite products, and the source has the required properties (finite products), it is left exact (by <a href="/functor-implication/left_exact_criterion">this result</a>).


structure_id: discrete_topology
property_id: fully faithful
type: functor
proof:
Since it is faithful and is full, it is fully faithful (by <a href="/functor-implication/fully-faithful-definition">this result</a>).


structure_id: discrete_topology
property_id: full on isomorphisms
type: functor
proof:
Since it is conservative and is full, it is full on isomorphisms (by <a href="/functor-implication/full-on-isomorphisms_criterion">this result</a>).


structure_id: discrete_topology
property_id: pseudomonic
type: functor
proof:
Since it is faithful and is full on isomorphisms, it is pseudomonic (by <a href="/functor-implication/pseudomonic_definition">this result</a>).


structure_id: enveloping_group
property_id: preserves equalizers
type: functor
proof:
Assume for contradiction that it preserves equalizers. Then it would preserve regular monomorphisms (by <a href="/functor-implication/equalizers_preservation_consequences">this result</a>) – contradiction.


structure_id: enveloping_group
property_id: preserves monomorphisms
type: functor
proof:
Assume for contradiction that it preserves monomorphisms. Then it would preserve monomorphisms, and the target has the required properties (mono-regular), so it would preserve regular monomorphisms (by <a href="/functor-implication/regular-mono-preserving_criterion">this result</a>) – contradiction.


structure_id: forget_group
property_id: conservative
type: functor
proof:
Since it is monadic, it is conservative (by <a href="/functor-implication/monadic_consequences">this result</a>).


structure_id: forget_group
property_id: right adjoint
type: functor
proof:
Since it is monadic, it is a right adjoint (by <a href="/functor-implication/monadic_consequences">this result</a>).


structure_id: forget_topology
property_id: right adjoint
type: functor
proof:
Since it is a coreflector, it is a right adjoint (by <a href="/functor-implication/dual_reflector_consequences">this result</a>).


structure_id: forget_topology
property_id: fully faithful
type: functor
proof:
Assume for contradiction that it is fully faithful. Then it would be conservative (by <a href="/functor-implication/conservative_criterion">this result</a>) – contradiction.


structure_id: forget_topology
property_id: left-invertible
type: functor
proof:
Assume for contradiction that it is left-invertible. Then it would be conservative (by <a href="/functor-implication/left-invertible_consequences">this result</a>) – contradiction.


structure_id: forget_topology
property_id: monadic
type: functor
proof:
Assume for contradiction that it is monadic. Then it would be conservative (by <a href="/functor-implication/monadic_consequences">this result</a>) – contradiction.


structure_id: forget_topology
property_id: comonadic
type: functor
proof:
Assume for contradiction that it is comonadic. Then it would be conservative (by <a href="/functor-implication/dual_monadic_consequences">this result</a>) – contradiction.


structure_id: forget_topology
property_id: full
type: functor
proof:
Assume for contradiction that it is full. Then it would be faithful and would be full, so it would be fully faithful (by <a href="/functor-implication/fully-faithful-definition">this result</a>) – contradiction.


structure_id: forget_abelian
property_id: preserves binary coproducts
type: functor
proof:
Assume for contradiction that it preserves binary coproducts. Then it would preserve binary coproducts and would preserve initial objects, and the source has the required properties (finite coproducts), so it would preserve finite coproducts (by <a href="/functor-implication/dual_preserve_finite_products_criterion">this result</a>) – contradiction.


structure_id: forget_abelian
property_id: right exact
type: functor
proof:
Assume for contradiction that it is right exact. Then it would preserve finite coproducts (by <a href="/functor-implication/dual_left_exact_consequences">this result</a>) – contradiction.


structure_id: forget_abelian
property_id: cocontinuous
type: functor
proof:
Assume for contradiction that it is cocontinuous. Then it would preserve coproducts (by <a href="/functor-implication/dual_continuous_consequences">this result</a>) – contradiction.


structure_id: forget_group_pointed
property_id: equivalence
type: functor
proof:
Assume for contradiction that it is an equivalence. Then it would be right-invertible (by <a href="/functor-implication/equivalence_invertible">this result</a>) – contradiction.


structure_id: free_group
property_id: faithful
type: functor
proof:
Since it is comonadic, it is faithful (by <a href="/functor-implication/dual_monadic_consequences">this result</a>).


structure_id: continuous_functions
property_id: regular
type: functor
proof:
Assume for contradiction that it is regular. Then it would preserve regular epimorphisms (by <a href="/functor-implication/regular_functor_definition">this result</a>) – contradiction.


structure_id: continuous_functions
property_id: right exact
type: functor
proof:
Assume for contradiction that it is right exact. Then it would preserve epimorphisms (by <a href="/functor-implication/dual_left_exact_consequences">this result</a>) – contradiction.


structure_id: continuous_functions
property_id: preserves coequalizers
type: functor
proof:
Assume for contradiction that it preserves coequalizers. Then it would preserve regular epimorphisms (by <a href="/functor-implication/dual_equalizers_preservation_consequences">this result</a>) – contradiction.


structure_id: continuous_functions
property_id: preserves reflexive coequalizers
type: functor
proof:
Assume for contradiction that it preserves reflexive coequalizers. Then it would preserve reflexive coequalizers, and the source has the required properties (pullbacks), so it would preserve regular epimorphisms (by <a href="/functor-implication/dual_another_regular-mono-preserving_criterion">this result</a>) – contradiction.


structure_id: pi_1
property_id: preserves epimorphisms
type: functor
proof:
Assume for contradiction that it preserves epimorphisms. Then it would preserve epimorphisms, and the target has the required properties (epi-regular), so it would preserve regular epimorphisms (by <a href="/functor-implication/dual_regular-mono-preserving_criterion">this result</a>) – contradiction.


structure_id: group_units
property_id: right-invertible
type: functor
proof:
Since it is a coreflector, it is right-invertible (by <a href="/functor-implication/dual_reflector_consequences">this result</a>).


structure_id: group_units
property_id: preserves terminal objects
type: functor
proof:
Since it preserves initial objects, and the source has the required properties (pointed), and the target has the required properties (pointed), it preserves terminal objects (by <a href="/functor-implication/dual_zero_preserving_condition">this result</a>).


structure_id: id_Set
property_id: equivalence
type: functor
proof:
Since it is an isomorphism, it is an equivalence (by <a href="/functor-implication/isomorphism_is_equivalence">this result</a>).


structure_id: id_Set
property_id: essentially surjective
type: functor
proof:
Since it is an equivalence, it is essentially surjective (by <a href="/functor-implication/equivalence_criterion">this result</a>).


structure_id: id_Set
property_id: fully faithful
type: functor
proof:
Since it is an equivalence, it is fully faithful (by <a href="/functor-implication/equivalence_criterion">this result</a>).


structure_id: id_Set
property_id: left-invertible
type: functor
proof:
Since it is an equivalence, it is left-invertible (by <a href="/functor-implication/equivalence_invertible">this result</a>).


structure_id: id_Set
property_id: right-invertible
type: functor
proof:
Since it is an equivalence, it is right-invertible (by <a href="/functor-implication/equivalence_invertible">this result</a>).


structure_id: id_Set
property_id: monadic
type: functor
proof:
Since it is an equivalence, it is monadic (by <a href="/functor-implication/equivalence_consequences">this result</a>).


structure_id: id_Set
property_id: reflector
type: functor
proof:
Since it is an equivalence, it is a reflector (by <a href="/functor-implication/equivalence_consequences">this result</a>).


structure_id: id_Set
property_id: comonadic
type: functor
proof:
Since it is an equivalence, it is comonadic (by <a href="/functor-implication/dual_equivalence_consequences">this result</a>).


structure_id: id_Set
property_id: coreflector
type: functor
proof:
Since it is an equivalence, it is a coreflector (by <a href="/functor-implication/dual_equivalence_consequences">this result</a>).


structure_id: id_Set
property_id: faithful
type: functor
proof:
Since it is fully faithful, it is faithful (by <a href="/functor-implication/fully-faithful-definition">this result</a>).


structure_id: id_Set
property_id: full
type: functor
proof:
Since it is fully faithful, it is full (by <a href="/functor-implication/fully-faithful-definition">this result</a>).


structure_id: id_Set
property_id: conservative
type: functor
proof:
Since it is fully faithful, it is conservative (by <a href="/functor-implication/conservative_criterion">this result</a>).


structure_id: modulo-p
property_id: preserves finite coproducts
type: functor
proof:
Since it preserves finite products, and the source has the required properties (biproducts), and the target has the required properties (biproducts), it preserves finite coproducts (by <a href="/functor-implication/biproduct_preserving_condition">this result</a>).


structure_id: modulo-p
property_id: preserves equalizers
type: functor
proof:
Assume for contradiction that it preserves equalizers. Then it would preserve equalizers and would preserve products, and the source has the required properties (products), so it would be continuous (by <a href="/functor-implication/continuous_criterion">this result</a>) – contradiction.


structure_id: monoid_ring
property_id: preserves regular monomorphisms
type: functor
proof:
Since it preserves coreflexive equalizers, and the source has the required properties (pushouts), it preserves regular monomorphisms (by <a href="/functor-implication/another_regular-mono-preserving_criterion">this result</a>).


structure_id: p-torsion
property_id: preserves coproducts
type: functor
proof:
Since it is finitary and preserves finite coproducts, and the source has the required properties (finite coproducts), it preserves coproducts (by <a href="/functor-implication/dual_product_criterion_filtered">this result</a>).


structure_id: p-torsion
property_id: preserves regular epimorphisms
type: functor
proof:
Assume for contradiction that it preserves regular epimorphisms. Then it would preserve regular epimorphisms, and the source has the required properties (epi-regular), so it would preserve epimorphisms (by <a href="/functor-implication/dual_mono-preserving_criterion">this result</a>) – contradiction.


structure_id: p-torsion
property_id: preserves coequalizers
type: functor
proof:
Assume for contradiction that it preserves coequalizers. Then it would preserve coequalizers and would preserve finite coproducts, and the source has the required properties (finite coproducts), so it would be right exact (by <a href="/functor-implication/dual_left_exact_criterion">this result</a>) – contradiction.


structure_id: pi_0
property_id: preserves coequalizers
type: functor
proof:
Assume for contradiction that it preserves coequalizers. Then it would preserve coequalizers and would preserve coproducts, and the source has the required properties (coproducts), so it would be cocontinuous (by <a href="/functor-implication/dual_continuous_criterion">this result</a>) – contradiction.


structure_id: pi_0
property_id: preserves reflexive coequalizers
type: functor
proof:
Assume for contradiction that it preserves reflexive coequalizers. Then it would preserve binary coproducts and would preserve reflexive coequalizers, and the source has the required properties (binary coproducts), so it would preserve coequalizers (by <a href="/functor-implication/dual_criterion_coreflexive_equalizers">this result</a>) – contradiction.


structure_id: trivial_groups
property_id: conservative
type: functor
proof:
Assume for contradiction that it is conservative. Then it would be conservative and would be full, so it would be full on isomorphisms (by <a href="/functor-implication/full-on-isomorphisms_criterion">this result</a>) – contradiction.


structure_id: trivial_groups
property_id: faithful
type: functor
proof:
Assume for contradiction that it is faithful. Then it would be faithful and would be full, so it would be fully faithful (by <a href="/functor-implication/fully-faithful-definition">this result</a>) – contradiction.


structure_id: walking_isomorphism_object_inclusion
property_id: equivalence
type: functor
proof:
Since it is essentially surjective and is fully faithful, it is an equivalence (by <a href="/functor-implication/equivalence_criterion">this result</a>).


structure_id: integer-rational-embedding
property_id: regular epimorphism
type: morphism
proof:
Assume for contradiction that it is a regular epimorphism. Then it would be a monomorphism and would be a regular epimorphism, so it would be an isomorphism (by <a href="/morphism-implication/regular_epi_and_mono_is_iso">this result</a>) – contradiction.


structure_id: integer-rational-embedding
property_id: regular monomorphism
type: morphism
proof:
Assume for contradiction that it is a regular monomorphism. Then it would be an epimorphism and would be a regular monomorphism, so it would be an isomorphism (by <a href="/morphism-implication/dual_regular_epi_and_mono_is_iso">this result</a>) – contradiction.


structure_id: integer-rational-embedding
property_id: split monomorphism
type: morphism
proof:
Assume for contradiction that it is a split monomorphism. Then it would be a regular monomorphism (by <a href="/morphism-implication/split_mono_is_regular_mono">this result</a>) – contradiction.


structure_id: integer-rational-embedding
property_id: split epimorphism
type: morphism
proof:
Assume for contradiction that it is a split epimorphism. Then it would be a regular epimorphism (by <a href="/morphism-implication/dual_split_mono_is_regular_mono">this result</a>) – contradiction.


structure_id: id_X
property_id: split monomorphism
type: morphism
proof:
Since it is an isomorphism, it is a split monomorphism (by <a href="/morphism-implication/iso_is_mono">this result</a>).


structure_id: id_X
property_id: split epimorphism
type: morphism
proof:
Since it is an isomorphism, it is a split epimorphism (by <a href="/morphism-implication/dual_iso_is_mono">this result</a>).


structure_id: id_X
property_id: regular monomorphism
type: morphism
proof:
Since it is a split monomorphism, it is a regular monomorphism (by <a href="/morphism-implication/split_mono_is_regular_mono">this result</a>).


structure_id: id_X
property_id: regular epimorphism
type: morphism
proof:
Since it is a split epimorphism, it is a regular epimorphism (by <a href="/morphism-implication/dual_split_mono_is_regular_mono">this result</a>).


structure_id: id_X
property_id: monomorphism
type: morphism
proof:
Since it is a regular monomorphism, it is a monomorphism (by <a href="/morphism-implication/regular_mono_is_mono">this result</a>).


structure_id: id_X
property_id: epimorphism
type: morphism
proof:
Since it is a regular epimorphism, it is an epimorphism (by <a href="/morphism-implication/dual_regular_mono_is_mono">this result</a>).


structure_id: empty-map
property_id: regular monomorphism
type: morphism
proof:
Since it is a monomorphism, and the category has the required properties (mono-regular), it is a regular monomorphism (by <a href="/morphism-implication/mono-regular_def">this result</a>).


structure_id: empty-map
property_id: isomorphism
type: morphism
proof:
Assume for contradiction that it is an isomorphism. Then it would be a split monomorphism (by <a href="/morphism-implication/iso_is_mono">this result</a>) – contradiction.


structure_id: empty-map
property_id: epimorphism
type: morphism
proof:
Assume for contradiction that it is an epimorphism. Then it would be an epimorphism and would be a monomorphism, and the category has the required properties (balanced), so it would be an isomorphism (by <a href="/morphism-implication/balanced_def">this result</a>) – contradiction.


structure_id: terminal-map
property_id: regular monomorphism
type: morphism
proof:
Assume for contradiction that it is a regular monomorphism. Then it would be a monomorphism (by <a href="/morphism-implication/regular_mono_is_mono">this result</a>) – contradiction.


