# nLab Grothendieck category

## Derived categories

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

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

## 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 small category

• 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.

• 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.

• 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 $X$

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

## Examples

• For $R$ a ring, $R$Mod is a Grothendieck category.

• For $C$ a small abelian category, the category $Ind(C)$ of ind-objects in $C$ is a Grothendieck category.

• for $C$ a Grothendieck category, the category $C_c$ of complexes in $C$ is again a Grothendieck category.

## References

Grothendieck categories are mentioned at the end of section 8.3 in

The relation to complexes is in section 14.1.

• 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)