abelian category

An abelian category is a category that behaves enough like the category of modules over a ring that all of homological algebra — kernels, cokernels, images, exact sequences, diagram chases — makes sense inside it. The motivating idea is to axiomatize exactly the features needed to do homology, without committing to actual modules, so that the same machinery applies to sheaves, chain complexes, and representations. It is the natural arena for derived functors, Ext, and Tor.

An abelian category is, in stages: a category with a zero object and with all finite products and coproducts that agree (so it is additive, and every Hom-set is an abelian group with bilinear composition); in which every morphism has a kernel and a cokernel; and crucially in which every monomorphism is the kernel of some morphism and every epimorphism is the cokernel of some morphism. This last normality axiom forces the first isomorphism theorem: every morphism factors as an epi onto its image followed by a mono, and image equals coimage.

The structural significance is captured by the Freyd-Mitchell embedding theorem: every small abelian category embeds, fully faithfully and exactly, into a category of modules R-Mod over some ring R. So abelian categories are exactly “module categories up to embedding,” which justifies the practice of proving diagram lemmas (the snake lemma, the five lemma) by element-chasing in modules and then invoking them in any abelian category. Standard examples include Ab, R-Mod, sheaves of abelian groups, and bounded chain complexes.