A Grothendieck category is an abelian category that has a generator, all small coproducts, and exact
filtered colimits. A generator is an object 
for which 
is faithful. Equivalently, a Grothendieck category
is an abelian category satisfying the AB5 axiom
and possessing a generator. Categories of modules and
categories of sheaves of modules on a ringed
space are basic examples. The Gabriel-Popescu
theorem represents every Grothendieck category as a localization
of a module category.
Grothendieck Category
See also
Abelian Category, Gabriel-Popescu TheoremExplore with Wolfram|Alpha
References
Popescu, N. Abelian Categories with Applications to Rings and Modules. London, England: Academic Press, 1973.Cite this as:
Weisstein, Eric W. "Grothendieck Category." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GrothendieckCategory.html