Implication Details
Assumptions: left cancellative
Conclusions: coreflexive equalizers, effective cocongruences, effective congruences, reflexive coequalizers
Proof: Any parallel pair of morphisms with a common section (or retraction) must be a pair of equal isomorphisms. In particular, they are the kernel pair of the identity morphism on the target, and the cokernel pair of the identity morphism on the source.
Show 43 categories using this implication
- category of algebras
- category of fields
- category of finite sets and bijections
- category of finite sets and injections
- category of locally ringed spaces
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of monoids
- category of non-empty sets
- category of pointed topological spaces
- category of posets
- category of prosets
- category of pseudo-metric spaces with non-expansive maps
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sets and relations
- category of sets with finite-to-one maps
- category of small categories
- category of smooth manifolds
- category of topological spaces
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category on two objects
- empty category
- poset [0,1]
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- simplex category
- trivial category
- walking coreflexive pair
- walking fork
- walking idempotent
- walking isomorphism
- walking parallel pair
- walking span