nLab
functoriality of categories of presheaves

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Category Theory

Contents

Idea

Given a small category CC, one can consider the category of presheaves PSh(C,D)PSh(C, D) valued in some category DD. Given some assumptions on DD, any functor of small categories F:CCF : C \to C' induces two adjoint pairs

F !:PSh(C,D)PSh(C,D):F * F_! : PSh(C, D) \rightleftarrows PSh(C', D) : F^*
F *:PSh(C,D)PSh(C,D):F * F^* : PSh(C', D) \rightleftarrows PSh(C, D) : F_*

Definitions

Definition

Let F:CCF : C \to C' be a functor of small categories and DD some category. The restriction of scalars functor F *:PSh(C,D)PSh(C,D)F^* : PSh(C', D) \to PSh(C, D) is given by the formula HHfH \mapsto H \circ f, i.e. mapping a presheaf H:C opDH : C'^{op} \to D to the composite

C opFC opHD.C^{op} \stackrel{F}{\to} C'^{op} \stackrel{H}{\to} D.
Proposition

Suppose DD admits small colimits (resp. small limits). Then the functor F *F^* admits a left adjoint F !F_! (resp. right adjoint F *F_*).

Properties

Lemma

Let F:CC:GF : C \rightleftarrows C' : G be an adjoint pair and consider the induced functors (F !,F *,F *)(F_!, F^*, F_*) and (G !,G *,G *)(G_!, G^*, G_*). One has

  • F !F_! is left adjoint to G !G_!,
  • F *F^* is left adjoint to G *G^*,
  • F *F_* is left adjoint to G *G_*,
  • F *=G *F_* = G^*,
  • F *=G !F^* = G_!.

Note that all these claims are in fact equivalent.

See also

For functoriality of sheaves, see

References

Last revised on July 24, 2014 at 01:22:04. See the history of this page for a list of all contributions to it.