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In mathematics, a '''Grothendieck category''' is a certain kind of [[abelian category]], introduced by [[Alexander Grothendieck]] in 1957<ref>{{citation|first=A.|last=Grothendieck|authorlink=Alexander Grothendieck|title=Sur quelques points d’algèbre homologique|journal=[[Tôhoku Mathematical Journal]]|volume=9|series=(2)|pages=119–221|year=1957|mr=0102537|url=http://projecteuclid.org/euclid.tmj/1178244839}}. [ftp://ftp.math.mcgill.ca/barr/pdffiles/gk.pdf English translation].</ref> in order to develop the machinery of [[homological algebra]] for [[module (mathematics)|module]]s and for [[sheaf (mathematics)|sheaves]] in a unified manner.

To every [[algebraic variety]] ''V'' one can associate a Grothendieck category qcoh(''V''), consisting of the [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''V''. This category encodes all the relevant information about ''V'', and ''V'' can be recovered from qcoh(''V''). This example gives rise to one approach to [[noncommutative algebraic geometry]]: the study of "non-commutative varieties" is then nothing but the study of Grothendieck categories.<ref>{{cite web|title=Quantum Ruled Surfaces|author=Izuru Mori|year=2007|url=http://mathsoc.jp/section/algebra/algsymp_past/algsymp07_files/mouri.pdf}}</ref>

==Definition==
By definition, a Grothendieck category ''A'' is an [[AB5 category]] with a [[Generator (category theory)|generator]]. Spelled out, this means that
* ''A'' is an [[abelian category]];
* every (possibly infinite) family of objects in ''A'' has a [[coproduct]] (a.k.a. direct sum) in ''A'';
* [[direct limit]]s (a.k.a. filtered colimits) of [[exact sequence]]s are exact; this means that if a direct system of [[short exact sequence]]s in ''A'' is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always [[exact functor|right-exact]]; the important point here is that we require them to be [[exact functor|left-exact]] as well.)
* ''A'' possesses a generator, i.e. there is an object ''G'' in ''A'' such that Hom(''G'',&nbsp;–) is a [[faithful functor]] from ''A'' to the [[category of sets]]. (In our situation, this is equivalent to saying that every object ''X'' of ''A'' admits an [[epimorphism]] ''G''<sup>(''I'')</sup>→''X'', where ''G''<sup>(''I'')</sup> denotes a direct sum of copies of ''G'', one for each element of the (possibly infinite) set ''I''.)

==Examples==
* The prototypical example of a Grothendieck category is the [[category of abelian groups]]; the group '''Z''' of integers can serve as a generator.
* More generally, given any [[ring (mathematics)|ring]] ''R'' (associative, with 1, but not necessarily commutative), the category Mod-''R'' of all left (or alternatively: right) [[module (mathematics)|module]]s over ''R'' is a Grothendieck category; ''R'' itself can serve as a generator.
* Given a [[topological space]] ''X'', the category of all [[sheaf (mathematics)|sheaves]] of abelian groups on ''X'' is a Grothendieck category. (More generally: the category of all sheaves of left ''R''-modules on ''X'' is a Grothendieck category for any ring ''R''.)  
* Given a [[ringed space]] (''X'',''O<sub>X</sub>''), the category of [[Sheaf_(mathematics)#Sheaves_of_modules|sheaves of ''O<sub>X</sub>''-modules]] is a Grothendieck category.
* Any category that's [[equivalence of categories|equivalent]] to a Grothendieck category is itself a Grothendieck category.
* Given a [[Category_(mathematics)#Small_and_large_categories|small category]] ''C'' and a Grothendieck category ''A'', the [[functor category]] Funct(''C'',''A'') is a Grothendieck category; if ''C'' is [[preadditive category|preadditive]], then the functor category Add(''C'',''A'') of all additive functors from ''C'' to ''A'' is a Grothendieck category as well.
* Given an (affine or projective) [[algebraic variety]] ''V'' (or more generally: a [[quasi-compact]] [[Glossary_of_scheme_theory#Separated_and_proper_morphisms|quasi-separated]] [[scheme (mathematics)|scheme]]), the category of [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''V'' is a Grothendieck category.

==Properties==
Every object in a Grothendieck category ''A'' has an [[injective hull]] in ''A''. This allows to construct [[injective resolution]]s and thereby the use of the tools of [[homological algebra]] in ''A''. Further, ''A'' contains an [[injective cogenerator]].

In a Grothendieck category, any family of [[subobject]]s (''U''<sub>''i''</sub>) of a given object ''X'' has a [[supremum]] Σ<sub>''i''</sub>''U''<sub>''i''</sub> which is again a subobject of ''X''. Further, if the family (''U''<sub>''i''</sub>) is directed (i.e. for any two objects in the family, the family has a third object that contains the two), and ''V'' is another subobject of ''X'', we have
:<math>\sum_{i}(U_i\cap V) = \left(\sum_{i}U_i\right) \cap V.</math>

In a Grothendieck category, arbitrary [[limit (category theory)|limit]]s (and in particular [[product (category theory)|product]]s) exist. It follows directly from the definition that arbitrary [[limit (category theory)|colimits]] and [[coproduct]]s (direct sums) exist as well. We can thus say that every Grothendieck category is [[complete category|complete and co-complete]]. Coproducts in a Grothendieck category are exact (i.e. the coproduct of a family of short exact sequences is again a short exact sequence), but products need not be exact.

If ''A'' is a Grothendieck category and ''T'' is a [[localizing subcategory]] of ''A'' (i.e. ''T'' is a full subcategory of ''A'', closed under ''A''-colimits, and for any [[short exact sequence]] <math>0\rarr A_1\rarr A_2\rarr A_3\rarr 0</math> in ''A'', ''A<sub>2</sub>'' is in ''T'' if and only if ''A<sub>1</sub>'' and ''A<sub>3</sub>'' are in ''T''), we can form the [[Serre quotient category]] ''A''/''T''. This quotient is again a Grothendieck category.

The [[Gabriel–Popesco theorem]] states that any Grothendieck category ''A'' is equivalent to a [[full subcategory]] of the category Mod-''R'' of right modules over some unital ring ''R'' (which can be taken to be the [[endomorphism ring]] of a generator of ''A''), and ''A'' can be obtained as a Serre quotient of Mod-''R'' by some localizing subcategory.<ref>{{cite journal |authors=N. Popesco, P. Gabriel|title=Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes|journal=Comptes Rendus de l'Académie des Sciences|volume=258|year=1964|pages=4188–4190}}</ref>

==References==
{{reflist}}
*{{cite book|author=N. Popescu|title=Abelian categories with applications to rings and modules|publisher=Academic Press|year=1973}}
*{{citation|first1=Pascual|last1=Jara|first2=Alain|last2=Verschoren|first3=Conchi|last3=Vidal|title=Localization and sheaves: a relative point of view|series=Pitman Research Notes in Mathematics Series|volume=339|publisher=Longman, Harlow|year=1995}}.

==External links==
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*[http://therisingsea.org/notes/AbelianCategories.pdf Abelian Categories], notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.

[[Category:Homological algebra]]
[[Category:Category theory]]