[[Category theory MOC]]
# Abelian category

An **abelian category** $\cat A$ is a [[preäbelian category]] such that every [[monomorphism]] is a [[Kernels and cokernels]] and every [[epimorphism]] is a [[Cokernel]]. #m/def/cat 
Thus in particular an abelian category is enriched over [[Category of abelian groups]] and admits finite [[Biproduct|biproducts]].

The prototypical example is [[Category of abelian groups]], or more generally [[Category of left modules]] for any [[ring]] $R$.
The [[Freyd-Mitchell theorem]] gives a sense in which _all_ abelian categories are categories of modules.

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