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 generalizes 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.