7.30 Localization of topoi
We repeat some of the material on localization to the apparently more general case of topoi. In reality this is not more general since we may always enlarge the underlying sites to assume that we are localizing at objects of the site.
Lemma 7.30.1. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is a topos. There is a canonical morphism of topoi
\[ j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
which is a localization as in Section 7.25 such that
the functor $j_\mathcal {F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H} \times \mathcal{F}/\mathcal{F}$, and
the functor $j_{\mathcal{F}!}$ is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$.
Proof.
Apply Lemma 7.29.5. This means we may assume $\mathcal{C}$ is a site with subcanonical topology, and $\mathcal{F} = h_ U = h_ U^\# $ for some $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Hence the material of Section 7.25 applies. In particular, there is an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ such that the composition
\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
is equal to $j_{U!}$, see Lemma 7.25.4. Denote $a : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$ the inverse functor, so $j_{\mathcal{F}!} = j_{U!} \circ a$, $j_\mathcal {F}^{-1} = a^{-1} \circ j_ U^{-1}$, and $j_{\mathcal{F}, *} = j_{U, *} \circ a$. The description of $j_{\mathcal{F}!}$ follows from the above. The description of $j_\mathcal {F}^{-1}$ follows from Lemma 7.25.7.
$\square$
Lemma 7.30.2. In the situation of Lemma 7.30.1, the functor $j_{\mathcal{F}, *}$ is the one associates to $\varphi : \mathcal{G} \to \mathcal{F}$ the sheaf
\[ U \longmapsto \{ \alpha : \mathcal{F}|_ U \to \mathcal{G}|_ U \text{ such that } \alpha \text{ is a right inverse to }\varphi |_ U \} . \]
Proof.
For any $\varphi : \mathcal{G} \to \mathcal{F}$, let us use the notation $\mathcal{G}/\mathcal{F}$ to denote the corresponding object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$. We have
\[ (j_{\mathcal{F}, *}(\mathcal{G}/\mathcal{F}))(U) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , j_{\mathcal{F}, *}(\mathcal{G}/\mathcal{F})) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}}(j_{\mathcal{F}}^{-1}h_ U^{\# }, (\mathcal{G}/\mathcal{F})). \]
By Lemma 7.30.1 this set is the fiber over the element $h_ U^\# \times \mathcal{F} \to \mathcal{F}$ under the map of sets
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# \times \mathcal{F}, \mathcal{G}) \xrightarrow {\varphi \circ } \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# \times \mathcal{F}, \mathcal{F}). \]
By the adjunction in Lemma 7.26.2, we have
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^{\# }\times \mathcal{F}, \mathcal{G}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^{\# },\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{G})) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U}), \\ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^{\# } \times \mathcal{F}, \mathcal{F}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^{\# },\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F},\mathcal{F})) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}), \end{align*}
and under the adjunction, the map $\varphi \circ $ becomes the map
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}),\quad \psi \longmapsto \varphi |_{\mathcal{C}/U} \circ \psi , \]
the element $h_ U^\# \times \mathcal{F} \to \mathcal{F}$ becomes $\text{id}_{\mathcal{F}|_{\mathcal{C}/U}}$. Therefore $(j_{\mathcal{F}, *}\mathcal{G}/\mathcal{F})(U)$ is isomorphic to the fiber of $\text{id}_{\mathcal{F}|_{\mathcal{C}/U}}$ under the map
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U}) \xrightarrow {\varphi |_{\mathcal{C}/U}\circ } \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}), \]
which is $\{ \alpha : \mathcal{F}|_ U \to \mathcal{G}|_ U \text{ such that } \alpha \text{ is a right inverse to }\varphi |_ U \} $ as desired.
$\square$
Lemma 7.30.3. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $\mathcal{C}/\mathcal{F}$ be the category of pairs $(U, s)$ where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $s \in \mathcal{F}(U)$. Let a covering in $\mathcal{C}/\mathcal{F}$ be a family $\{ (U_ i, s_ i) \to (U, s)\} $ such that $\{ U_ i \to U\} $ is a covering of $\mathcal{C}$. Then $j : \mathcal{C}/\mathcal{F} \to \mathcal{C}$ is a continuous and cocontinuous functor of sites which induces a morphism of topoi $j : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{F}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. In fact, there is an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{F}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ which turns $j$ into $j_\mathcal {F}$.
Proof.
We omit the verification that $\mathcal{C}/\mathcal{F}$ is a site and that $j$ is continuous and cocontinuous. By Lemma 7.21.5 there exists a morphism of topoi $j$ as indicated, with $j^{-1}\mathcal{G}(U, s) = \mathcal{G}(U)$, and there is a left adjoint $j_!$ to $j^{-1}$. A morphism $\varphi : * \to j^{-1}\mathcal{G}$ on $\mathcal{C}/\mathcal{F}$ is the same thing as a rule which assigns to every pair $(U, s)$ a section $\varphi (s) \in \mathcal{G}(U)$ compatible with restriction maps. Hence this is the same thing as a morphism $\varphi : \mathcal{F} \to \mathcal{G}$ over $\mathcal{C}$. We conclude that $j_!* = \mathcal{F}$. In particular, for every $\mathcal{H} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{F})$ there is a canonical map
\[ j_!\mathcal{H} \to j_!* = \mathcal{F} \]
i.e., we obtain a functor $j'_! : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{F}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$. An inverse to this functor is the rule which assigns to an object $\varphi : \mathcal{G} \to \mathcal{F}$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ the sheaf
\[ a(\mathcal{G}/\mathcal{F}) : (U, s) \longmapsto \{ t \in \mathcal{G}(U) \mid \varphi (t) = s\} \]
We omit the verification that $a(\mathcal{G}/\mathcal{F})$ is a sheaf and that $a$ is inverse to $j'_!$.
$\square$
Definition 7.30.4. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$.
The topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is called the localization of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ at $\mathcal{F}$.
The morphism of topoi $j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of Lemma 7.30.1 is called the localization morphism.
We are going to show that whenever the sheaf $\mathcal{F}$ is equal to $h_ U^\# $ for some object $U$ of the site, then the localization of the topos is equal to the category of sheaves on the localization of the site at $U$. Moreover, we are going to check that any functorialities are compatible with this identification.
Lemma 7.30.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F} = h_ U^\# $ for some object $U$ of $\mathcal{C}$. Then $j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ constructed in Lemma 7.30.1 agrees with the morphism of topoi $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ constructed in Section 7.25 via the identification $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ of Lemma 7.25.4.
Proof.
We have seen in Lemma 7.25.4 that the composition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is $j_{U!}$. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is $j_{\mathcal{F}!}$ by Lemma 7.30.1. Hence $j_{\mathcal{F}!} = j_{U!}$ via the identification. So $j_\mathcal {F}^{-1} = j_ U^{-1}$ (by adjointness) and so $j_{\mathcal{F}, *} = j_{U, *}$ (by adjointness again).
$\square$
Lemma 7.30.6. Let $\mathcal{C}$ be a site. If $s : \mathcal{G} \to \mathcal{F}$ is a morphism of sheaves on $\mathcal{C}$ then there exists a natural commutative diagram of morphisms of topoi
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{G} \ar[rd]_{j_\mathcal {G}} \ar[rr]_ j & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[ld]^{j_\mathcal {F}} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) & } \]
where $j = j_{\mathcal{G}/\mathcal{F}}$ is the localization of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ at the object $\mathcal{G}/\mathcal{F}$. In particular we have
\[ j^{-1}(\mathcal{H} \to \mathcal{F}) = (\mathcal{H} \times _\mathcal {F} \mathcal{G} \to \mathcal{G}) \]
and
\[ j_!(\mathcal{E} \xrightarrow {e} \mathcal{F}) = (\mathcal{E} \xrightarrow {s \circ e} \mathcal{G}). \]
Proof.
The description of $j^{-1}$ and $j_!$ comes from the description of those functors in Lemma 7.30.1. The equality of functors $j_{\mathcal{G}!} = j_{\mathcal{F}!} \circ j_!$ is clear from the description of these functors (as forgetful functors). By adjointness we also obtain the equalities $j_\mathcal {G}^{-1} = j^{-1} \circ j_\mathcal {F}^{-1}$, and $j_{\mathcal{G}, *} = j_{\mathcal{F}, *} \circ j_*$.
$\square$
Lemma 7.30.7. Assume $\mathcal{C}$ and $s : \mathcal{G} \to \mathcal{F}$ are as in Lemma 7.30.6. If $\mathcal{G} = h_ V^\# $ and $\mathcal{F} = h_ U^\# $ and $s : \mathcal{G} \to \mathcal{F}$ comes from a morphism $V \to U$ of $\mathcal{C}$ then the diagram in Lemma 7.30.6 is identified with diagram (7.25.8.1) via the identifications $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ V^\# $ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ of Lemma 7.25.4.
Proof.
This is true because the descriptions of $j^{-1}$ agree. See Lemma 7.25.9 and Lemma 7.30.6.
$\square$
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