In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957[1] 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.
This category encodes all the relevant geometric information about
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.
Spelled out, this means that The name "Grothendieck category" appeared neither in Grothendieck's Tôhoku paper[1] nor in Gabriel's thesis;[2] 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.)
[1][2] 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.)
, we have[6] 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).
is complete,[7] i.e. that arbitrary limits (and in particular products) exist in
is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in
This follows from Peter J. Freyd's special adjoint functor theorem and its dual.
[9] As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.
of left-exact additive (covariant) functors
in a Grothendieck category is called finitely generated if, whenever
is written as the sum of a family of subobjects of
is finitely generated if, and only if, for any directed system
[11] A Grothendieck category need not contain any non-zero finitely generated objects.
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.
Again, this generalizes the notion of finitely presented modules.
In a locally finitely generated Grothendieck category
, the finitely presented objects can be characterized as follows:[12]
is finitely presented if, and only if, for every directed system
[13] (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.
in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence
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.