Alexander Grothendieck

Alexander Grothendieck

Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

 has introduced (in his famous Tohoku article) a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according to the axiom (AB2). AB3 categories are abelian categories

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

 possessing arbitrary coproducts (hence by existence of quotients in abelian categories also all colimits). AB5 categories are the AB3 categories in which filtered colimits of exact sequences are exact. Grothendieck categories

Grothendieck category

In mathematics, a Grothendieck category is an AB5 category with a generator. In other words, it is an abelian category A admitting arbitrary coproducts, for which filtered colimits of exact sequences are exact and which possess a generator, i.e...

are the AB5 categories with a generator.