# nLab Grothendieck category

### Context

#### Additive and abelian categories

additive and abelian categories

## Derived categories

#### Homological algebra

homological algebra

Introduction

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 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.

• 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

• 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.)

• 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.

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

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

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 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,

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)

Last revised on June 11, 2017 at 09:35:16. See the history of this page for a list of all contributions to it.