# nLab Grothendieck category

## 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

• 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 $X$;

• 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)