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7.22 Cocontinuous functors which have a right adjoint

It may happen that a cocontinuous functor $u$ has a right adjoint $v$. In this case it is often the case that $v$ is continuous, and if so, then it defines a morphism of topoi (which is the same as the one 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 $g_*\mathcal{F}$ is equal to the presheaf $v^ p\mathcal{F}$, in other words, $(g_*\mathcal{F})(V) = \mathcal{F}(v(V))$.

Proof. We have $u^ ph_ V = h_{v(V)}$ by Lemma 7.19.3. By Lemma 7.20.4 this implies that $g^{-1}(h_ V^\# ) = (u^ ph_ V^\# )^\# = (u^ ph_ V)^\# = h_{v(V)}^\# $. Hence for any sheaf $\mathcal{F}$ on $\mathcal{C}$ we have

\begin{eqnarray*} (g_*\mathcal{F})(V) & = & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h_ V^\# , g_*\mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(g^{-1}(h_ V^\# ), \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_{v(V)}^\# , \mathcal{F}) \\ & = & \mathcal{F}(v(V)) \end{eqnarray*}

which proves the lemma. $\square$

In the situation of Lemma 7.22.1 we see that $v^ p$ transforms sheaves into sheaves. Hence we can define $v^ s = v^ p$ restricted to sheaves. Just as in Lemma 7.13.3 we see that $v_ s : \mathcal{G} \mapsto (v_ p\mathcal{G})^\# $ is a left adjoint to $v^ s$. On the other hand, we have $v^ s = g_*$ and $g^{-1}$ is a left adjoint of $g_*$ as well. We conclude that $g^{-1} = v_ s$ is exact.

Lemma 7.22.2. In the situation of Lemma 7.22.1. We have $g_* = v^ s = v^ p$ and $g^{-1} = v_ s = (v_ p\ )^\# $. If $v$ is continuous then $v$ defines a morphism of sites $f$ from $\mathcal{C}$ to $\mathcal{D}$ whose associated morphism of topoi is equal to the morphism $g$ associated to the cocontinuous functor $u$. In other words, a continuous functor which has a cocontinuous left adjoint defines a morphism of sites.

Proof. Clear from the discussion above the lemma and Definitions 7.14.1 and Lemma 7.15.2. $\square$

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 $u$.

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