functoriality of categories of presheaves

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

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

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

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

$C^{op} \stackrel{F}{\to} C'^{op} \stackrel{H}{\to} D.$

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

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

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

Note that all these claims are in fact equivalent.

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

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

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

$(F^* F_! y I) J \cong (F^* y F I) J = (y F I) (F J) = Hom(F J, F I) \cong Hom(J, I) = (y I) J.$

For functoriality of sheaves, see

- SGA 4, Exp. I, no. 5.
- Stacks Project, tags 00VC and 00XF.

Last revised on January 29, 2020 at 04:54:24. See the history of this page for a list of all contributions to it.