(also nonabelian homological algebra)
additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
There is a sequence of extra structure and property on a category that makes this category behave like a general context for homological algebra. In order of increasing structure and property this is:
Ab-enriched category: a category that is Ab-enriched;
pre-additive category: an $Ab$-enriched category that has a terminal object or initial object and therefore a zero object; Notice however that many authors (e.g. Weibel, Popescu) by preadditive (or pre-additive) simply mean $Ab$-enriched.
additive category: a pre-additive category that admits binary products or binary coproducts and therefore binary biproducts (equivalently, an $Ab$-enriched category with all finite products or coproducts);
pre-abelian category: an additive category that admits kernels and cokernels (equivalently, an $Ab$-enriched category with all finite limits and colimits);
abelian category: a pre-abelian category such that every monomorphism is a kernel and every epimorphism is a cokernel (and many other equivalent definitions).
Pre-abelian and abelian categories are sometimes called (AB1) and (AB2) categories, after the sequence of additional axioms on top of additive categories introduced by Grothendieck in Tohoku. AB1 and AB2 are self-dual axioms (AB1 is existence of kernels and cokernels, and AB2 requires that, for any $f$, the canonical morphism $\mathrm{Coim}\,f\to \mathrm{Im}\,f$ is an isomorphism). These continue in non-selfdual manner:
AB3: an abelian category with all coproducts (hence with all colimits);
AB4: an (AB3) category in which coproducts of monomorphisms are monic;
AB5: an (AB3) category in which filtered colimits of exact sequences exist and are exact;
AB6: an (AB3) category such that
The concepts (AB3–AB6) also have dual forms (AB3–AB6).
There are further refinements along these lines. In particular
Various further axiom structures are considered for additive (sometimes abelian) categories.
Grothendieck category: an AB5-category with a generator
Gabriel’s property sup
Various generic classes of examples of additive and abelian categories are of relevance: