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
Here, $F^*$ is given by precomposition with $F$, whereas $F_!$ and $F_*$ are the left and right (global) Kan extensions along $F$.
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
($P_\odot$ and $P^\odot$ are not standard notations) where $P_\odot$ is the Yoneda extension of $P$.
A functor $F : C \to C'$ gives rise to two profunctors:
a companion profunctor $\hat F : C \nrightarrow C'$, given by
After currying, this amounts to the functor $y \circ F : C \to PSh(C')$.
a conjoint profunctor $\check F : C' \nrightarrow C$, given by
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 $F_* = (F^* \circ y)^\odot$.
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
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
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
For functoriality of sheaves, see
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.
Last revised on February 16, 2024 at 13:49:22. See the history of this page for a list of all contributions to it.