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.
- 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.)
- 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: a quasi-compact quasi-separated 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 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 , such as 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).
An object in a Grothendieck category is called finitely generated if the sum of every directed family of proper subobjects of is again a proper subobject of . (In the case of module categories, this notion is equivalent to the familiar notion of finitely generated modules.) 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. In such a category, every object is the sum of its finitely generated subobjects.
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 categories 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.
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 .
- Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal, (2), 9 (2): 119–221, doi:10.2748/tmj/1178244839, MR 0102537. English translation.
- Gabriel, Pierre (1962), "Des catégories abéliennes" (PDF), Bull. Soc. Math. Fr., 90: 323–448
- Izuru Mori (2007). "Quantum Ruled Surfaces" (PDF).
- Faith, Carl (1973). Algebra: Rings, Modules and Categories I. Springer. pp. 486–498. ISBN 9783642806346.
- Mac Lane, Saunders (1978). Categories for the Working Mathematician, 2nd edition. Springer. p. 130.
- Popesco, Nicolae; Gabriel, Pierre (1964). "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes". Comptes rendus de l'Académie des Sciences. 258: 4188–4190.
- Šťovíček, Jan (2013-01-01). "Deconstructibility and the Hill Lemma in Grothendieck categories". Forum Mathematicum. 25 (1). arXiv:1005.3251. Bibcode:2010arXiv1005.3251S. doi:10.1515/FORM.2011.113.
- Popescu, Nicolae (1973). Abelian categories with applications to rings and modules. Academic Press.
- Jara, Pascual; Verschoren, Alain; Vidal, Conchi (1995), Localization and sheaves: a relative point of view, Pitman Research Notes in Mathematics Series, 339, Longman, Harlow.