nLab functoriality of categories of presheaves

Idea

Functoriality w.r.t. functors

Given a small category $C$, one can consider the category of presheaves $PSh(C, D)$ valued in some category $D$. Given that $D$ possesses small limits and colimits, any functor of small categories $F \colon C \to C'$ induces two adjoint pairs

$$F_! \,\colon\, PSh(C, D) \rightleftarrows PSh(C', D) \,\colon\, F^\ast$$

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

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

Functoriality w.r.t. profunctors

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

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

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

A functor $F \,\colon\, C \to C'$ gives rise to two profunctors:

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

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

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

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

  $$\check F \,\colon\, C' \times {C}^{op} \to D \,\colon\, (c', c) \mapsto Hom_{C'}(Fc, c').$$

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

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

Definitions

Definition

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

Properties

Lemma

Let $F \,\colon\, C \rightleftarrows C' \,\colon\, G$ be an adjoint pair and consider the induced functors $(F_!, F^\ast, F_\ast)$ 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. The bind operation for $PSh$ is given by Yoneda extension and the Kleisli 2-category of $PSh$ is Prof.

References

• SGA 4, Exp. I, no. 5.
• Stacks Project, tags 00VC and 00XF.
• Andreas Nuyts, Lifting Profunctors to Presheaf Categories, note