nLab Grothendieck category



Additive and abelian categories

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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

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


In terms of the ABnn 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

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).


A Grothendieck category CC satisfies the following properties.

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



(RRMod is Grothendieck abelian)
For RR a commutative ring, its category of modules RRMod 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)].


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


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

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


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


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

The relation to complexes is in section 14.1.

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

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

Proof that RRMod internal to any Grothendieck topos is Grothendieck abelian:

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:

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

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):

On Grothendieck abelian categories of chain complexes:

which was published as

Last revised on May 13, 2024 at 09:47:32. See the history of this page for a list of all contributions to it.