nLab functoriality of categories of presheaves



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Functoriality w.r.t. functors

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_*

Here, F *F^* is given by precomposition with FF, whereas F !F_! and F *F_* are the left and right (global) Kan extensions along FF.

Functoriality w.r.t. profunctors

In fact, more generally, any profunctor 𝒫:CC\mathcal{P} : C \nrightarrow C' (i.e. 𝒫:C×C opD\mathcal{P} : C \times {C'}^{op} \to D or, after currying, P:CPSh(C,D)P : C \to PSh(C', D)) gives rise to a single adjoint pair

P :PSh(C,D)PSh(C,D):P P_\odot : PSh(C, D) \rightleftarrows PSh(C', D) : P^\odot

(P P_\odot and P P^\odot are not standard notations) where P P_\odot is the Yoneda extension of PP.

A functor F:CCF : C \to C' gives rise to two profunctors:

  • a companion profunctor F^:CC\hat F : C \nrightarrow C', given by

    F^:C×C opD:(c,c)Hom C(c,Fc). \hat F : C \times {C'}^{op} \to D : (c, c') \mapsto Hom_{C'}(c', Fc).

    After currying, this amounts to the functor yF:CPSh(C)y \circ F : C \to PSh(C').

  • a conjoint profunctor Fˇ:CC\check F : C' \nrightarrow C, given by

    Fˇ:C×C opD:(c,c)Hom C(Fc,c). \check F : C' \times {C}^{op} \to D : (c', c) \mapsto Hom_{C'}(Fc, c').

    After currying, this amounts to the functor F *y:CPSh(C)F^* \circ y : C' \to PSh(C).

The former profunctor produces the adjoint pair F !F *F_! \dashv F^*, i.e. F !=(yF) F_! = (y \circ F)_\odot and F *(yF) F^* \cong (y \circ F)^\odot. The latter profunctor produces the adjoint pair F *F *F^* \dashv F_*, i.e. F *(F *y) F^* \cong (F^* \circ y)_\odot and F *=(F *y) F_* = (F^* \circ y)^\odot.



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.

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



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_* \cong G^*,
  • F *G !F^* \cong G_!.

Note that all these claims are in fact equivalent.


If FF is fully faithful, then so are F !F_! and F *F_*.


A left adjoint functor LRL \dashv R is fully faithful precisely if RLR L is naturally isomorphic to the identity functor (by the unit). Dually, RR is fully faithful precisely if LRL R is naturally isomorphic to the identity (by the co-unit). Hence, it suffices to prove F *F !IdF^* F_! \cong Id, which by uniqueness of the right adjoint immediately implies that F *F *IdF^* F_* \cong Id and thus proves both claims.

Being a left adjoint, F *F !F^* F_! preserves colimits. Because every presheaf is a colimit of representable objects, it is sufficient to show that F *F !yyF^* F_! y \cong y where yy is the Yoneda-embedding. We have

(F *F !yI)J(F *yFI)J=(yFI)(FJ)=Hom(FJ,FI)Hom(J,I)=(yI)J. (F^* F_! y I) J \cong (F^* y F I) J = (y F I) (F J) = Hom(F J, F I) \cong Hom(J, I) = (y I) J.

See also

For functoriality of sheaves, see

The pseudofunctor PShPSh (with !-_! as its action on morphisms) takes a category to its free cocompletion. As such, it has the structure of a weak 2-monad, and more specifically it is a prototypical example of a lax-idempotent 2-monad. It factors over its Eilenberg-Moore 2-category, the 2-category of (total? cocomplete?) categories, as a lax-idempotent 2-adjunction. The bind operation for PShPSh is given by Yoneda extension and the Kleisli 2-category of PShPSh is Prof.


Last revised on February 16, 2024 at 13:49:22. See the history of this page for a list of all contributions to it.