nLab functoriality of categories of presheaves

Idea

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

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

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

Definitions

Definition

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

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

Proposition

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

Properties

Lemma

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 $G_*$,
• $F_* \cong G^*$,
• $F^* \cong G_!$.

Note that all these claims are in fact equivalent.

Lemma

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

Proof

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

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

See also

For functoriality of sheaves, see

• restriction and extension of sheaves
• direct image
• inverse image

The pseudofunctor PSh (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.

References

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