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 1962 thesis.
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 GabrielâÂÂRosenberg 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.
By definition, a Grothendieck category is an AB5 category with a generator. Spelled out, this means that
The name "Grothendieck category" appeared neither 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, not requiring the existence of a generator.)
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 GabrielâÂÂPopescu 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 GabrielâÂÂPopescu, 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 .
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.
A Grothendieck category is locally finitely presented if it has a set of finitely presented generators, which is equivalent to saying that every object is a direct limit of finitely presented objects.
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.
A Grothendieck category is called spectral if every short exact sequence splits. To a given Grothendieck category one can define a spectral Grothendieck category and a canonical functor . The objects of correspond, up to isomorphism, to the injective objects of . The functor turns all essential monomorphisms into isomorphisms and thus identifies each object of with its injective hull. This allows to define useful dimension-like functions on : for each indecomposable injective object in and every object in , the cardinal number measures, roughly speaking, how many direct summands isomorphic to ' appear in the injective hull of .