nLab
Grothendieck category

Context

Additive and abelian categories

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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

Definition

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 small category

  • that admits a generator;

  • that admits small colimits;

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

    • for II a directed set and 0A iB iC i00 \to A_i \to B_i \to C_i \to 0 an exact sequence for each iIi \in I, then 0colim iA icolim iB icolim iC i00 \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 CC satisfies the following properties.

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

Examples

References

Grothendieck categories are mentioned at the end of section 8.3 in

The relation to complexes is in section 14.1.

See also books

  • Peter Freyd, Abelian categories, Harper (1966O)
  • Nicolae Popescu, An introduction to Abelian categories with applications to rings and modules, Academic Press 1973

The duality of Grothendieck categories with categories of modules over linearly compact ring?s is discussed in

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

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)

Revised on March 6, 2014 08:53:29 by Zoran Škoda (161.53.130.104)