Editing Abelian category (section)

==Grothendieck's axioms==
In his [[Tōhoku article]], Grothendieck listed four additional axioms (and their duals) that an abelian category '''A''' might satisfy. These axioms are still in common use to this day. They are the following:
* AB3) For every indexed family (''A''<sub>''i''</sub>) of objects of '''A''', the [[coproduct]] *''A''<sub>i</sub> exists in '''A''' (i.e. '''A''' is [[cocomplete]]).
* AB4) '''A''' satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.
* [[AB5 category|AB5]]) '''A''' satisfies AB3), and [[filtered colimit]]s of [[exact sequence]]s are exact.
and their duals
* AB3*) For every indexed family (''A''<sub>''i''</sub>) of objects of '''A''', the [[Product (category theory)|product]] P''A''<sub>''i''</sub> exists in '''A''' (i.e. '''A''' is [[Complete category|complete]]).
* AB4*) '''A''' satisfies AB3*), and the product of a family of epimorphisms is an epimorphism.
* AB5*) '''A''' satisfies AB3*), and [[filtered limit]]s of exact sequences are exact.

Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically:
* AB1) Every morphism has a kernel and a cokernel.
* AB2) For every morphism ''f'', the canonical morphism from coim ''f'' to im ''f'' is an [[isomorphism]].

Grothendieck also gave axioms AB6) and AB6*).

* AB6) '''A''' satisfies AB3), and given a family of filtered categories <math>I_j, j\in J</math> and maps <math>A_j : I_j \to A</math>, we have <math>\prod_{j\in J} \lim_{I_j} A_j = \lim_{I_j, \forall j\in J} \prod_{j\in J} A_j</math>, where lim denotes the filtered colimit.
* AB6*) '''A''' satisfies AB3*), and given a family of cofiltered categories <math>I_j, j\in J</math> and maps <math>A_j : I_j \to A</math>, we have <math>\sum_{j\in J} \lim_{I_j} A_j = \lim_{I_j, \forall j\in J} \sum_{j\in J} A_j</math>, where lim denotes the cofiltered limit.