In mathematics, a Grothendieck category (named after 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...

) is an AB5 category

AB5 category

Alexander Grothendieck has introduced 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...

 with a generator

Generator (category theory)

In category theory in mathematics a generator of a category \mathcal C is an object G of the category, such that for any two different morphisms f, g: X \rightarrow Y in \mathcal C, there is a morphism h : G \rightarrow X, such that the compositions f \circ h \neq g \circ h.Generators are central...

. In other words, it is an abelian category

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...

 A admitting arbitrary coproduct

Coproduct

In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

s, for which filtered colimits of exact sequences are exact and which possess a generator, i.e. an object E of A such that Hom(E, –) is a faithful functor from A to the category of sets. A dual concept is a coGrothendieck category.

External links

• Grothendieck category in nLab