7.22 Cocontinuous functors which have a right adjoint
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of the underlying categories such that $v$ is right adjoint to $u$. In this case, if $v$ is continuous, then $u$ is cocontinuous (Lemma 7.22.4). If $u$ is cocontinuous, then it is often (but not always, see Example 7.22.5) the case that $v$ is continuous, and if so, then $v$ defines a morphism of sites whose associated morphism of topoi is the same as that defined by $u$.
Lemma 7.22.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$, and $v : \mathcal{D} \to \mathcal{C}$ be functors. Assume that $u$ is cocontinuous and that $v$ is a right adjoint to $u$. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated to $u$, see Lemma 7.21.1. Then
for a sheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf $g_*\mathcal{F}$ is equal to the presheaf $v^ p\mathcal{F}$, in other words, $(g_*\mathcal{F})(V) = \mathcal{F}(v(V))$, and
for a sheaf $\mathcal{G}$ on $\mathcal{D}$ we have $g^{-1}\mathcal{G} = (v_ p\mathcal{G})^\# $.
Proof.
For $\mathcal{F}$ as in (1) we have
\[ g_*\mathcal{F} = {}_ su\mathcal{F} = {}_ pu\mathcal{F} = v^ p\mathcal{F} = \mathcal{F} \circ v \]
The first equality is Lemma 7.21.1. The second equality is Lemma 7.20.2. The third equality is Lemma 7.19.3. The final equality is the definition of $v^ p$ in Section 7.5. This proves (1). For $\mathcal{G}$ as in (2) we have
\[ g^{-1}\mathcal{G} = (u^ p\mathcal{G})^\# = (v_ p\mathcal{G})^\# \]
The first equality is Lemma 7.21.1. The second equality is Lemma 7.19.3.
$\square$
Lemma 7.22.2. Notation and assumptions as in Lemma 7.22.1. If in addition $v$ is continuous then $v$ defines a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ whose associated morphism of topoi is equal to $g$.
Proof.
We will use the results of Lemma 7.22.1 without further mention. To prove that $v$ defines a morphism of sites $f$ as in the statement of the lemma, we have to show that $v_ s$ is an exact functor (see Definition 7.14.1). Since $v_ s\mathcal{G} = (v_ p\mathcal{G})^\# = g^{-1}\mathcal{G}$ this follows from the fact that $g$ is a morphism of topoi. Then we see that $f^{-1} = v_ s = g^{-1}$ and we find that $f = g$ as morphisms of topoi.
$\square$
Example 7.22.3. This example continues the discussion of Example 7.14.3 from which we borrow the notation $\mathcal{C}, \tau , \tau ', \epsilon $. Observe that the identity functor $v : \mathcal{C}_{\tau '} \to \mathcal{C}_\tau $ is a continuous functor and the identity functor $u : \mathcal{C}_\tau \to \mathcal{C}_{\tau '}$ is a cocontinuous functor. Moreover $u$ is left adjoint to $v$. Hence the results of Lemmas 7.22.1 and 7.22.2 apply and we conclude $v$ defines a morphism of sites, namely
\[ \epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau '} \]
whose corresponding morphism of topoi is the same as the morphism of topoi associated to the cocontinuous functor $u$.
Lemma 7.22.4. Let $\mathcal{C}$ and $\mathcal{D}$ be sites and let $v : \mathcal{D} \to \mathcal{C}$ be a continuous functor. Assume $v$ has a left adjoint $u : \mathcal{C} \to \mathcal{D}$. Then
$u$ is cocontinuous,
the results of Lemmas 7.22.1 and 7.22.2 hold.
In particular, $v$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$.
Proof.
Let $U$ be an object of $\mathcal{C}$ and let $\{ V_ j \to u(U)\} _{j \in J}$ be a covering in $\mathcal{D}$. Then $\{ v(V_ j) \to v(u(U))\} _{i \in I}$ is a covering in $\mathcal{C}$. Via the adjunction map $U \to v(u(U))$ we can base change this to a covering $\{ W_ j \to U\} _{j \in J}$ with $W_ j = v(V_ j) \times _{v(u(U))} U$. Denoting $p_ j : W_ j \to v(V_ j)$ the first projection, we obtain maps
\[ u(W_ j) \xrightarrow {u(p_ j)} u(v(V_ j)) \longrightarrow V_ j \]
where the second arrow is the adjunction map. This determines a morphism $\{ u(W_ j) \to u(U)\} \to \{ V_ j \to u(U)\} $ of families of maps with fixed target, showing that $u$ is indeed cocontinuous. The other statements are immediate from Lemmas 7.22.1 and 7.22.2.
$\square$
Example 7.22.5. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of the underlying categories such that $v$ is right adjoint to $u$. Lemma 7.22.4 shows that if $v$ is continuous, then $u$ is cocontinous. Conversely, if $u$ is cocontinuous, then we can't conclude that $v$ is continuous. We will give an example of this phenomenon using the big étale and smooth sites of a scheme, but presumably there is an elementary example as well. Namely, consider a scheme $S$ and the sites $(\mathit{Sch}/S)_{\acute{e}tale}$ and $(\mathit{Sch}/S)_{smooth}$. We may assume these sites have the same underlying category, see Topologies, Remark 34.11.1. Let $u = v = \text{id}$. Then $u$ as a functor from $(\mathit{Sch}/S)_{\acute{e}tale}$ to $(\mathit{Sch}/S)_{smooth}$ is cocontinuous as every smooth covering of a scheme can be refined by an étale covering, see More on Morphisms, Lemma 37.38.7. Conversely, the functor $v$ from $(\mathit{Sch}/S)_{smooth}$ to $(\mathit{Sch}/S)_{\acute{e}tale}$ is not continuous as a smooth covering is not an étale covering in general.
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