Grothendieck category

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.
To every algebraic variety formula_1 one can associate a Grothendieck category formula_2, consisting of the quasi-coherent sheaves on formula_1. This category encodes all the relevant geometric information about formula_1, and formula_1 can be recovered from formula_2 (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories.
Definition.
By definition, a Grothendieck category formula_7 is an AB5 category with a generator. Spelled out, this means that
The name "Grothendieck category" neither appeared in Grothendieck's Tôhoku paper nor in Gabriel's thesis; it came into use in the second half of the 1960s in the work of several authors, including Jan-Erik Roos, Bo Stenström, Ulrich Oberst, and Bodo Pareigis. (Some authors use a different definition in that they don't require the existence of a generator.)
Properties and theorems.
Every Grothendieck category contains an injective cogenerator. For example, an injective cogenerator of the category of abelian groups is the quotient group formula_55.
Every object in a Grothendieck category formula_7 has an injective hull in formula_7. This allows to construct injective resolutions and thereby the use of the tools of homological algebra in formula_7, in order to define derived functors. (Note that not all Grothendieck categories allow projective resolutions for all objects; examples are categories of sheaves of abelian groups on many topological spaces, such as on the space of real numbers.)
In a Grothendieck category, any family of subobjects formula_59 of a given object formula_17 has a supremum (or "sum") formula_61 as well as an infimum (or "intersection") formula_62, both of which are again subobjects of formula_17. Further, if the family formula_59 is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and formula_1 is another subobject of formula_17, we have
Grothendieck categories are well-powered (sometimes called "locally small", although that term is also used for a different concept), i.e. the collection of subobjects of any given object forms a set (rather than a proper class).
It is a rather deep result that every Grothendieck category formula_7 is complete, i.e. that arbitrary limits (and in particular products) exist in formula_7. By contrast, it follows directly from the definition that formula_7 is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in formula_7. 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.
A functor formula_72 from a Grothendieck category formula_7 to an arbitrary category formula_74 has a left adjoint if and only if it commutes with all limits, and it has a right adjoint if and only if it commutes with all colimits. This follows from Peter J. Freyd's "special adjoint functor theorem" and its dual.
The Gabriel–Popescu theorem states that any Grothendieck category formula_7 is equivalent to a full subcategory of the category formula_26 of right modules over some unital ring formula_24 (which can be taken to be the endomorphism ring of a generator of formula_7), and formula_7 can be obtained as a Gabriel quotient of formula_26 by some localizing subcategory.
As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable. Furthermore, Gabriel-Popescu can be used to see that every Grothendieck category is complete, being a reflective subcategory of the complete category formula_26 for some formula_24. 
Every small abelian category formula_40 can be embedded in a Grothendieck category, in the following fashion. The category formula_84 of left-exact additive (covariant) functors formula_85 (where formula_86 denotes the category of abelian groups) is a Grothendieck category, and the functor formula_87, with formula_88, is full, faithful and exact. A generator of formula_7 is given by the coproduct of all formula_90, with formula_91. The category formula_7 is equivalent to the category formula_93 of ind-objects of formula_40 and the embedding formula_95 corresponds to the natural embedding formula_96. We may therefore view formula_7 as the co-completion of formula_40.
Special kinds of objects and Grothendieck categories.
An object formula_17 in a Grothendieck category is called "finitely generated" if, whenever formula_17 is written as the sum of a family of subobjects of formula_17, then it is already the sum of a finite subfamily. (In the case formula_102 of module categories, this notion is equivalent to the familiar notion of finitely generated modules.) Epimorphic images of finitely generated objects are again finitely generated. If formula_103 and both formula_104 and formula_105 are finitely generated, then so is formula_17. The object formula_17 is finitely generated if, and only if, for any directed system formula_108 in formula_109 in which each morphism is a monomorphism, the natural morphism formula_110 is an isomorphism. A Grothendieck category need not contain any non-zero finitely generated objects. 
A Grothendieck category is called "locally finitely generated" if it has a set of finitely generated generators (i.e. if there exists a family formula_111 of finitely generated objects such that to every object formula_17 there exist formula_113 and a non-zero morphism formula_114; equivalently: formula_17 is epimorphic image of a direct sum of copies of the formula_116). In such a category, every object is the sum of its finitely generated subobjects. Every category formula_102 is locally finitely generated.
An object formula_17 in a Grothendieck category is called "finitely presented" if it is finitely generated and if every epimorphism formula_119 with finitely generated domain formula_120 has a finitely generated kernel. Again, this generalizes the notion of finitely presented modules. If formula_103 and both formula_104 and formula_105 are finitely presented, then so is formula_17. In a locally finitely generated Grothendieck category formula_109, the finitely presented objects can be characterized as follows: formula_17 in formula_109 is finitely presented if, and only if, for every directed system formula_108 in formula_109, the natural morphism formula_110 is an isomorphism.
An object formula_17 in a Grothendieck category formula_109 is called "coherent" if it is finitely presented and if each of its finitely generated subobjects is also finitely presented. (This generalizes the notion of coherent sheaves on a ringed space.) The full subcategory of all coherent objects in formula_109 is abelian and the inclusion functor is exact.
An object formula_17 in a Grothendieck category is called "Noetherian" if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence formula_135 of subobjects of formula_17 eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. (In the case formula_102, this notion is equivalent to the familiar notion of Noetherian modules.) A Grothendieck category is called "locally Noetherian" if it has a set of Noetherian generators; an example is the category of left modules over a left-Noetherian ring.