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Grothendieck category

Idea

Grothendieck categories are those abelian categories $𝒜$

• such that for presheaves on a site with values in $𝒜$ there is an existence theorem for the sheafification functor;

• such that all complexes in $𝒜$ (with respect to a translation) are quasi-isomorphic to homotopically injective complexes (so that derived functor can be computed on homotopically injective replacements).

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

• 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 $0\to {\mathrm{colim}}_i{A}_i\to {\mathrm{colim}}_i{B}_i\to {\mathrm{colim}}_i{C}_i\to 0$ 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 $F:C^{\mathrm{op}}\to \mathrm{Set}$ commutes with small limits, the $F$ is representable;

• if a functor $F:C^{\mathrm{op}}\to \mathrm{Set}$ commutes with small colimits, then $F$ has a right adjoint.

• If $C$ is equipped with translation $T:C\to C$, then for every complex $X\in \mathrm{Cplx}(C)$ 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 $\mathrm{Ind}(C)$ 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.

Related concepts

• sheaf of abelian groups

• abelian sheaf cohomology

References

Grothendieck categories are mentioned at the end of section 8.3 in

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves

The relation to complexes is in section 14.1.

See also the book

• Peter Freyd, Abelian categories, Harper (1966O)

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.