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  one can associate a Grothendieck category , consisting of the quasi-coherent sheaves on . This category encodes all the relevant geometric information about , and  can be recovered from  (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  is an AB5 category with a generator. Spelled out, this means that
  is an abelian category;
 every (possibly infinite) family of objects in  has a coproduct (also known as direct sum) in ;
 direct limits of short exact sequences are exact; this means that if a direct system of short exact sequences in  is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always right-exact; the important point here is that we require them to be left-exact as well.) possesses a generator, i.e. there is an object  in  such that  is a faithful functor from  to the category of sets. (In our situation, this is equivalent to saying that every object  of  admits an epimorphism , where  denotes a direct sum of copies of , one for each element of the (possibly infinite) set .) 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.) Examples
 The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group  of integers can serve as a generator. More generally, given any ring  (associative, with , but not necessarily commutative), the category  of all right (or alternatively: left) modules over  is a Grothendieck category;  itself can serve as a generator. Given a topological space , the category of all sheaves of abelian groups on  is a Grothendieck category. (More generally: the category of all sheaves of right -modules on  is a Grothendieck category for any ring .) Given a ringed space , the category of sheaves of OX-modules is a Grothendieck category. Given an (affine or projective) algebraic variety  (or more generally: any scheme), the category  of quasi-coherent sheaves on  is a Grothendieck category. Given a small site (C, J) (i.e. a small category C together with a Grothendieck topology J), the category of all sheaves of abelian groups on the site is a Grothendieck category. Constructing further Grothendieck categories 

 Any category that's equivalent to a Grothendieck category is itself a Grothendieck category. Given Grothendieck categories  , the product category   is a Grothendieck category. Given a small category  and a Grothendieck category , the functor category , consisting of all covariant functors from  to , is a Grothendieck category. Given a small preadditive category  and a Grothendieck category , the functor category  of all additive covariant functors from  to  is a Grothendieck category. If  is a Grothendieck category and  is a localizing subcategory of , then both  and the Serre quotient category  are Grothendieck categories. 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 . Every object in a Grothendieck category  has an injective hull in . This allows to construct injective resolutions and thereby the use of the tools of homological algebra in , 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  of a given object  has a supremum (or "sum")  as well as an infimum (or "intersection") , both of which are again subobjects of . Further, if the family  is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and  is another subobject of , 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  is complete, i.e. that arbitrary limits (and in particular products) exist in . By contrast, it follows directly from the definition that  is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in . 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  from a Grothendieck category  to an arbitrary category  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  is equivalent to a full subcategory of the category  of right modules over some unital ring  (which can be taken to be the endomorphism ring of a generator of ), and  can be obtained as a Gabriel quotient of  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  for some . Every small abelian category  can be embedded in a Grothendieck category, in the following fashion. The category  of left-exact additive (covariant) functors  (where  denotes the category of abelian groups) is a Grothendieck category, and the functor , with , is full, faithful and exact. A generator of  is given by the coproduct of all , with . The category  is equivalent to the category  of ind-objects of  and the embedding  corresponds to the natural embedding  . We may therefore view  as the co-completion of . Special kinds of objects and Grothendieck categories 
An object  in a Grothendieck category is called finitely generated if, whenever  is written as the sum of a family of subobjects of , then it is already the sum of a finite subfamily. (In the case  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  and both  and  are finitely generated, then so is . The object  is finitely generated if, and only if, for any directed system  in  in which each morphism is a monomorphism, the natural morphism  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  of finitely generated objects such that to every object  there exist  and a non-zero morphism ; equivalently:  is epimorphic image of a direct sum of copies of the ). In such a category, every object is the sum of its finitely generated subobjects. Every category  is locally finitely generated. An object  in a Grothendieck category is called finitely presented if it is finitely generated and if every epimorphism  with finitely generated domain  has a finitely generated kernel. Again, this generalizes the notion of finitely presented modules. If  and both  and  are finitely presented, then so is . In a locally finitely generated Grothendieck category , the finitely presented objects can be characterized as follows:  in  is finitely presented if, and only if, for every directed system  in , the natural morphism  is an isomorphism. An object  in a Grothendieck category  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  is abelian and the inclusion functor is exact. An object  in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence  of subobjects of  eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. (In the case , 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. Notes

References

External links

Abelian Categories, notes by Daniel Murfet. Section 2.3 covers Grothendieck categories. Category theory
Additive categories
Homological algebra