# Equivalent definitions of abelian categories - reference request

Etingof et. al. define abelian categories as additive categories in which for every morphism $$\phi : X \to Y$$ there exists a sequence $$K \xrightarrow{k} X \xrightarrow{i} I \xrightarrow{j} Y \xrightarrow{c} C$$ with the following properties

1. $$ji = \phi$$,
2. $$(K,k) = \ker \phi, (C,c) = \operatorname{coker} \phi$$,
3. $$(I,i) = \operatorname{coker} k, (I,j) = \ker c$$.

Why is this equivalent to the standard definition of an abelian category: an additive category is abelian if

1. every map has kernels and cokernels,

2. every monomorphism is the kernel of its cokernel,

3. every epimorphism is the cokernel of its kernel.

I assume this is a standard result in some book/paper but I can't find it anywhere.

• The term to look up is "image factorization." Jun 12, 2017 at 17:42
• Related (possibly a duplicate) : math.stackexchange.com/questions/45008/… Jun 12, 2017 at 20:35

This sequence is basically giving us the image factorisation of $\phi$, and I'll assume that you've seen that you can always construct this factorisation in an abelian category.
• Every morphism $\phi$ has a kernel and cokernel, because we need them to define the above sequence.
• Suppose $\phi$ is a monomorphism. Its kernel is the zero morphism, so the cokernel of that kernel is isomorphic (in the suitable sense) to the identity. Therefore $\phi$ is isomorphic in the same sense to $j$, the kernel of the cokernel. Thus $\phi$ is also the kernel of its cokernel.