Grothendieck category

Summary

A Grothendieck category is an abelian category with all small colimits, in which filtered colimits are exact, and which has a generator. Grothendieck introduced these conditions in his 1957 Tôhoku paper and proved that such categories have enough injective objects. This made it possible to define sheaf cohomology as a derived functor. Module categories and categories of sheaves of abelian groups are Grothendieck categories. By the Gabriel–Popescu theorem, every Grothendieck category is a localization of a module category.