Implication Details

Assumptions: cokernels, kernels, preadditive

Conclusions: quotients of congruences

Proof: For any congruence $E$ on an object $X$ of a preadditive category, let $E_0$ be the kernel of $p_2 : E \to X$ . The restriction of $p_1$ to $E_0$ is a monomorphism. We can then see that $E$ must be the pullback of $p_1 - p_2 : E \to X$ and $E_0 \hookrightarrow X$ . Then the cokernel of $E_0 \hookrightarrow X$ is a quotient of $E$ .

Show 11 categories using this implication

category of abelian groups
category of abelian sheaves
category of filtered vector spaces
category of finite abelian groups
category of finite-dimensional vector spaces [countable field]
category of finite-dimensional vector spaces [finite field]
category of finite-dimensional vector spaces [uncountable field]
category of finitely generated abelian groups
category of left modules over a division ring
category of left modules over a ring
category of vector spaces