7.19 More functoriality of presheaves

In this section we revisit the material of Section 7.5. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. Recall that

$u^ p : \textit{PSh}(\mathcal{D}) \longrightarrow \textit{PSh}(\mathcal{C})$

is the functor that associates to $\mathcal{G}$ on $\mathcal{D}$ the presheaf $u^ p\mathcal{G} = \mathcal{G} \circ u$. It turns out that this functor not only has a left adjoint (namely $u_ p$) but also a right adjoint.

Namely, for any $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ we define a category ${}_ V\mathcal{I} = {}_ V^ u\mathcal{I}$. Its objects are pairs $(U, \psi : u(U) \to V)$. Note that the arrow is in the opposite direction from the arrow we used in defining the category $\mathcal{I}_ V^ u$ in Section 7.5. A morphism $(U, \psi ) \to (U', \psi ')$ is given by a morphism $\alpha : U \to U'$ such that $\psi = \psi ' \circ u(\alpha )$. In addition, given any presheaf of sets $\mathcal{F}$ on $\mathcal{C}$ we introduce the functor ${}_ V\mathcal{F} : {}_ V\mathcal{I}^{opp} \to \textit{Sets}$, which is defined by the rule ${}_ V\mathcal{F}(U, \psi ) = \mathcal{F}(U)$. We define

${}_ pu(\mathcal{F})(V) := \mathop{\mathrm{lim}}\nolimits _{{}_ V\mathcal{I}^{opp}} {}_ V\mathcal{F}$

As a limit there are projection maps $c(\psi ) : {}_ pu(\mathcal{F})(V) \to \mathcal{F}(U)$ for every object $(U, \psi )$ of ${}_ V\mathcal{I}$. In fact,

${}_ pu(\mathcal{F})(V) = \left\{ \begin{matrix} \text{collections } s_{(U, \psi )} \in \mathcal{F}(U) \\ \forall \beta : (U_1, \psi _1) \to (U_2, \psi _2) \text{ in }{}_ V\mathcal{I} \\ \text{ we have } \beta ^*s_{(U_2, \psi _2)} = s_{(U_1, \psi _1)} \end{matrix} \right\}$

where the correspondence is given by $s \mapsto s_{(U, \psi )} = c(\psi )(s)$. We leave it to the reader to define the restriction mappings ${}_ pu(\mathcal{F})(V) \to {}_ pu(\mathcal{F})(V')$ associated to any morphism $V' \to V$ of $\mathcal{D}$. The resulting presheaf will be denoted ${}_ pu\mathcal{F}$.

Lemma 7.19.1. There is a canonical map ${}_ pu\mathcal{F}(u(U)) \to \mathcal{F}(U)$, which is compatible with restriction maps.

Proof. This is just the projection map $c(\text{id}_{u(U)})$ above. $\square$

Note that any map of presheaves $\mathcal{F} \to \mathcal{F}'$ gives rise to compatible systems of maps between functors ${}_ V\mathcal{F} \to {}_ V\mathcal{F}'$, and hence to a map of presheaves ${}_ pu\mathcal{F} \to {}_ pu\mathcal{F}'$. In other words, we have defined a functor

${}_ pu : \textit{PSh}(\mathcal{C}) \longrightarrow \textit{PSh}(\mathcal{D})$

Lemma 7.19.2. The functor ${}_ pu$ is a right adjoint to the functor $u^ p$. In other words the formula

$\mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ p\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{D})}(\mathcal{G}, {}_ pu\mathcal{F})$

holds bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

Proof. This is proved in exactly the same way as the proof of Lemma 7.5.4. We note that the map $u^ p{}_ pu \mathcal{F} \to \mathcal{F}$ from Lemma 7.19.1 is the map that is used to go from the right to the left.

Alternately, think of a presheaf of sets $\mathcal{F}$ on $\mathcal{C}$ as a presheaf $\mathcal{F}'$ on $\mathcal{C}^{opp}$ with values in $\textit{Sets}^{opp}$, and similarly on $\mathcal{D}$. Check that $({}_ pu \mathcal{F})' = u_ p(\mathcal{F}')$, and that $(u^ p\mathcal{G})' = u^ p(\mathcal{G}')$. By Remark 7.5.5 we have the adjointness of $u_ p$ and $u^ p$ for presheaves with values in $\textit{Sets}^{opp}$. The result then follows formally from this. $\square$

Thus given a functor $u : \mathcal{C} \to \mathcal{D}$ of categories we obtain a sequence of functors

$u_ p, u^ p, {}_ pu$

between categories of presheaves where in each consecutive pair the first is left adjoint to the second.

Lemma 7.19.3. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of categories. Assume that $v$ is right adjoint to $u$. Then we have

1. $u^ ph_ V = h_{v(V)}$ for any $V$ in $\mathcal{D}$,

2. the category $\mathcal{I}^ v_ U$ has an initial object,

3. the category ${}_ V^ u\mathcal{I}$ has a final object,

4. ${}_ pu = v^ p$, and

5. $u^ p = v_ p$.

Proof. Proof of (1). Let $V$ be an object of $\mathcal{D}$. We have $u^ ph_ V = h_{v(V)}$ because $u^ ph_ V(U) = \mathop{Mor}\nolimits _\mathcal {D}(u(U), V) = \mathop{Mor}\nolimits _\mathcal {C}(U, v(V))$ by assumption.

Proof of (2). Let $U$ be an object of $\mathcal{C}$. Let $\eta : U \to v(u(U))$ be the map adjoint to the map $\text{id} : u(U) \to u(U)$. Then we claim $(u(U), \eta )$ is an initial object of $\mathcal{I}_ U^ v$. Namely, given an object $(V, \phi : U \to v(V))$ of $\mathcal{I}_ U^ v$ the morphism $\phi$ is adjoint to a map $\psi : u(U) \to V$ which then defines a morphism $(u(U), \eta ) \to (V, \phi )$.

Proof of (3). Let $V$ be an object of $\mathcal{D}$. Let $\xi : u(v(V)) \to V$ be the map adjoint to the map $\text{id} : v(V) \to v(V)$. Then we claim $(v(V), \xi )$ is a final object of ${}_ V^ u\mathcal{I}$. Namely, given an object $(U, \psi : u(U) \to V)$ of ${}_ V^ u\mathcal{I}$ the morphism $\psi$ is adjoint to a map $\phi : U \to v(V)$ which then defines a morphism $(U, \psi ) \to (v(V), \xi )$.

Hence for any presheaf $\mathcal{F}$ on $\mathcal{C}$ we have

\begin{eqnarray*} v^ p\mathcal{F}(V) & = & \mathcal{F}(v(V)) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{v(V)}, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ ph_ V, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{D})}(h_ V, {}_ pu\mathcal{F}) \\ & = & {}_ pu\mathcal{F}(V) \end{eqnarray*}

which proves part (4). Part (5) follows by the uniqueness of adjoint functors. $\square$

Lemma 7.19.4. A continuous functor of sites which has a continuous left adjoint defines a morphism of sites.

Proof. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $w : \mathcal{D} \to \mathcal{C}$ be a continuous left adjoint. Then $u_ p = w^ p$ by Lemma 7.19.3. Hence $u_ s = w^ s$ has a left adjoint, namely $w_ s$ (Lemma 7.13.3). Thus $u_ s$ has both a right and a left adjoint, whence is exact (Categories, Lemma 4.24.6). $\square$

Comment #4065 by Frid on

In the last line of the proof of Lemma 7.19.3., it should say "part (4)" and "Part (5)" instead of "part (2)" and "Part (3)".

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