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Tag: 00XR

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Lemma 7.19.5. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that
1. $u$ is cocontinuous, and
2. $u$ is continuous.
Let $g : \mathop{\textit{Sh}}\nolimits(\mathcal{C}) \to \mathop{\textit{Sh}}\nolimits(\mathcal{D})$ be the associated morphism of topoi. Then
1. sheafification in the formula $g^{-1} = (u^p )^\#$ is unnecessary, in other words $g^{-1}(\mathcal{G})(U) = \mathcal{G}(u(U))$,
2. $g^{-1}$ has a left adjoint $g_{!} = (u_p )^\#$, and
3. $g^{-1}$ commutes with arbitrary limits and colimits.

Proof. By Lemma 7.13.2 for any sheaf $\mathcal{G}$ on $\mathcal{D}$ the presheaf $u^p\mathcal{G}$ is a sheaf on $\mathcal{C}$. And then we see the adjointness by the following string of equalities \begin{eqnarray*} \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G}) & = & \mathop{\rm Mor}\nolimits_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G}) \\ & = & \mathop{\rm Mor}\nolimits_{\textit{PSh}(\mathcal{D})}(u_p\mathcal{F}, \mathcal{G}) \\ & = & \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(\mathcal{D})}(g_{!}\mathcal{F}, \mathcal{G}) \end{eqnarray*} The statement on limits and colimits follows from the discussion in Categories, Section 4.23. $\square$

\begin{lemma}
\label{lemma-when-shriek}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor.
Assume that
\begin{enumerate}
\item[(a)] $u$ is cocontinuous, and
\item[(b)] $u$ is continuous.
\end{enumerate}
Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$
be the associated morphism of topoi. Then
\begin{enumerate}
\item sheafification in the formula $g^{-1} = (u^p\ )^\#$ is
unnecessary, in other words $g^{-1}(\mathcal{G})(U) = \mathcal{G}(u(U))$,
\item $g^{-1}$ has a left adjoint $g_{!} = (u_p\ )^\#$, and
\item $g^{-1}$ commutes with arbitrary limits and colimits.
\end{enumerate}
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-pushforward-sheaf} for any sheaf $\mathcal{G}$
on $\mathcal{D}$ the presheaf $u^p\mathcal{G}$ is a sheaf on $\mathcal{C}$.
And then we see the adjointness by the following string of
equalities
\begin{eqnarray*}
\Mor_{\Sh(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G})
& = &
\Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G})
\\
& = &
\Mor_{\textit{PSh}(\mathcal{D})}(u_p\mathcal{F}, \mathcal{G})
\\
& = &
\Mor_{\Sh(\mathcal{D})}(g_{!}\mathcal{F}, \mathcal{G})
\end{eqnarray*}
The statement on limits and colimits follows from the
\end{proof}


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