nLab Grothendieck category

Idea

Grothendieck categories are those abelian categories $\mathcal{A}$

• such that for presheaves on a site with values in $\mathcal{A}$ there is an existence theorem for the sheafification functor;

• such that all complexes in $\mathcal{A}$ (with respect to a translation) are quasi-isomorphic to homotopically injective complexes (so that derived functor can be computed on homotopically injective replacements).

More abstractly, Grothendieck categories are precisely Ab-enriched Grothendieck toposes. This follows from the Gabriel-Popescu theorem together with the theory of enriched sheaves.

Definition

In terms of the AB$n$ hierarchy discussed at additive and abelian categories we have

A Grothendieck category is an AB5-category which has a generator.

This means that a Grothendieck category is an abelian category

• that admits a generator;

• that admits small colimits;

• such that small filtered colimits are exact in the following sense:

  • for $I$ a directed set and $0 \to A_i \to B_i \to C_i \to 0$ an exact sequence for each $i \in I$, then $0 \to colim_i A_i \to colim_i B_i \to colim_i C_i \to 0$ is also an exact sequence.

Dually a co-Grothendieck category is an AB5$^*$ category with a cogenerator. The category of abelian groups is not a co-Grothendieck category. Any abelian category which is simultaneously Grothendieck and co-Grothendieck has just a single object (see Freyd’s book, p.116).

Properties

A Grothendieck category $C$ satisfies the following properties.

• it admits small limits;

• if a functor $F : C^{op} \to Set$ commutes with small limits, the $F$ is representable;

• if a functor $F : C^{op} \to Set$ commutes with small colimits, then $F$ has a right adjoint.

• The Gabriel-Popescu theorem exhibits any Grothendieck abelian category as a reflective subcategory of a category of modules over a ring.

• Any Grothendieck abelian category is locally presentable.

  (Beke (2000), Prop. 3.10, see also Krause (2015), Corollary 5.2 and references therein.)

• More generally, any cocomplete abelian category with a set of generators in which $\kappa$-filtered colimits commute with finite limits for some regular cardinal $\kappa$ is a locally presentable category. See Positselski-Rosicky, Theorem 2.2.

• If $C$ is equipped with translation $T : C \to C$, then for every complex $X \in Cplx(C)$ there exists a quasi-isomorphism of complexes $X \to I$ such that $I$ is homotopically injective.

• it satisfies Pierre Gabriel’s sup property: every small family of subobjects of a given object $X$ has a supremum which is a subobject of $X$;

• it admits an injective cogenerator (see Kashiwara-Schapira, Theorem 9.6.3).

Much of the localization theory of rings generalizes to general Grothendieck categories.

Examples

Example

($R$Mod is Grothendieck abelian)
For $R$ a commutative ring, its category of modules $R$Mod is a Grothendieck category. (See e.g Kiersz 06, prop. 4 for the proof that filtered colimits here are exact.) This statement remains true internal to any Grothendieck topos [Johnstone (1977), Thm. 8.11 (iii)].

Example

(Vect is Grothendieck abelian)
Taking $R$ in Exp. to be a field it follows that categories of vector spaces are Gorthendieck abelian.

Example

($Ch_\bullet(\mathcal{A}_{Gr})$ is Grothendieck abelian)
For $\mathcal{A}$ a Grothendieck category, the (unbounded) category of chain complexes $Ch_\bullet(\mathcal{A})$ in $\mathcal{A}$ is again a Grothendieck category (e.g. Hovey (1999), p. 3).

With Exp. it follows that the usual categories $Ch_\bullet(R Mod)$ of chain complexes of modules over a ring are all Grothendieck abelian.

Example

($Ind(\mathcal{A})$ is Grothendieck abelian)
For $\mathcal{A}$ a small abelian category, the category $Ind(\mathcal{A})$ of ind-objects in $C$ is a Grothendieck category.

Related concepts

• sheaf of abelian groups

• abelian sheaf cohomology

References

A dedicated survey is

• Grigory Garkusha, Grothendieck categories, Algebra i Analiz 2001, 55 pp. pdf

Grothendieck categories are mentioned at the end of section 8.3 in

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves

The relation to complexes is in section 14.1.

The proof that filtered colimits in $R Mod$ are exact is spelled out for instance in

• Andy Kiersz, Colimits and homological algebra, 2006 (pdf)

Proof that $R$Mod internal to any Grothendieck topos is Grothendieck abelian:

• Peter Johnstone: Theorem 8.11 in: Topos Theory, Academic Press (1977) Dover reprint (2014)

See also:

• Peter Freyd, Abelian categories, Harper (1966O)

• Nicolae Popescu, An introduction to Abelian categories with applications to rings and modules, Academic Press 1973

The fact that all Grothendieck categories are locally presentable:

• Tibor Beke, Prop. 3.10 in: Sheafifiable homotopy model categories, Math. Proc. Cambridge Philosophical Society 129 3 (2000) 447-475 [arXiv:math/0102087, doi:10.1017/S0305004100004722]

• Henning Krause, Corollary 5.2 in: Deriving Auslander’s formula, Documenta Math. 20 (2015) 669-688 [arXiv:1409.7051]

A generalization to $\kappa$-Grothendieck categories (defined using $\kappa$-filtered colimits) is proved in Theorem 2.2 of

• Leonid Positselski, Jiří Rosický, Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories, arXiv:1512.08119.

The duality of Grothendieck categories with categories of modules over linearly compact rings is discussed in

• U. Oberst, Duality theory for Grothendieck categories and linearly compact rings, J. Algebra 15 (1970), p. 473 –542, pdf

Discussion of model structures on chain complexes in Grothendieck abelian categories is in

• Denis-Charles Cisinski, F. Déglise, Local and stable homologial algebra in Grothendieck abelian categories, Homology, Homotopy and Applications, vol. 11 (1) (2009) (pdf)

Formalization of Grothendieck categories as univalent categories in homotopy type theory: Formalization of abelian univalent categories of ring-modules, in homotopy type theory (univalent foundations of mathematics):

• Jarl G. Taxerås Flaten, Section 3.1 in Univalent categories of modules [arXiv:2207.03261]

On Grothendieck abelian categories of chain complexes:

• Mark Hovey, p. 3 of: Model category structures on chain complexes of sheaves (1999) [K-theory:0366, pdf, pdf]

which was published as

• Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 6 (2001) [ams:S0002-9947-01-02721-0, jstor:221954]