7.25 Localization
Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. See Categories, Example 4.2.13 for the definition of the category $\mathcal{C}/U$ of objects over $U$. We turn $\mathcal{C}/U$ into a site by declaring a family of morphisms $\{ V_ j \to V\} $ of objects over $U$ to be a covering of $\mathcal{C}/U$ if and only if it is a covering in $\mathcal{C}$. Consider the forgetful functor
\[ j_ U : \mathcal{C}/U \longrightarrow \mathcal{C}. \]
This is clearly cocontinuous and continuous. Hence by the results of the previous sections we obtain a morphism of topoi
\[ j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
given by $j_ U^{-1}$ and $j_{U*}$, as well as a functor $j_{U!}$.
Definition 7.25.1. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.
The site $\mathcal{C}/U$ is called the localization of the site $\mathcal{C}$ at the object $U$.
The morphism of topoi $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is called the localization morphism.
The functor $j_{U*}$ is called the direct image functor.
For a sheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_ U^{-1}\mathcal{F}$ is called the restriction of $\mathcal{F}$ to $\mathcal{C}/U$.
For a sheaf $\mathcal{G}$ on $\mathcal{C}/U$ the sheaf $j_{U!}\mathcal{G}$ is called the extension of $\mathcal{G}$ by the empty set.
The restriction $j_ U^{-1}\mathcal{F}$ is the sheaf defined by the rule $j_ U^{-1}\mathcal{F}(X/U) = \mathcal{F}(X)$ as expected. The extension by the empty set also has a very easy description in this case; here it is.
Lemma 7.25.2. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}/U$. Then $j_{U!}(\mathcal{G}^\# )$ is the sheaf associated to the presheaf
\[ V \longmapsto \coprod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U) \]
with obvious restriction mappings.
Proof.
By Lemma 7.21.5 we have $j_{U!}(\mathcal{G}^\# ) = ((j_ U)_ p\mathcal{G}^\# )^\# $. By Lemma 7.13.4 this is equal to $((j_ U)_ p\mathcal{G})^\# $. Hence it suffices to prove that $(j_ U)_ p$ is given by the formula above for any presheaf $\mathcal{G}$ on $\mathcal{C}/U$. OK, and by the definition in Section 7.5 we have
\[ (j_ U)_ p\mathcal{G}(V) = \mathop{\mathrm{colim}}\nolimits _{(W/U, V \to W)} \mathcal{G}(W) \]
Now it is clear that the category of pairs $(W/U, V \to W)$ has an object $O_\varphi = (\varphi : V \to U, \text{id} : V \to V)$ for every $\varphi : V \to U$, and moreover for any object there is a unique morphism from one of the $O_\varphi $ into it. The result follows.
$\square$
Lemma 7.25.3. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $X/U$ be an object of $\mathcal{C}/U$. Then we have $j_{U!}(h_{X/U}^\# ) = h_ X^\# $.
Proof.
Denote $p : X \to U$ the structure morphism of $X$. By Lemma 7.25.2 we see $j_{U!}(h_{X/U}^\# )$ is the sheaf associated to the presheaf
\[ V \longmapsto \coprod \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \{ \psi : V \to X \mid p \circ \psi = \varphi \} \]
This is clearly the same thing as $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, X)$. Hence the lemma follows.
$\square$
We have $j_{U!}(*) = h_ U^\# $ by either of the two lemmas above. Hence for every sheaf $\mathcal{G}$ over $\mathcal{C}/U$ there is a canonical map of sheaves $j_{U!}\mathcal{G} \to h_ U^\# $. This characterizes sheaves in the essential image of $j_{U!}$.
Lemma 7.25.4. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ gives an equivalence of categories
\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \]
Proof.
Let us denote objects of $\mathcal{C}/U$ as pairs $(X, a)$ where $X$ is an object of $\mathcal{C}$ and $a : X \to U$ is a morphism of $\mathcal{C}$. Similarly, objects of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ are pairs $(\mathcal{F}, \varphi )$. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ sends $\mathcal{G}$ to the pair $(j_{U!}\mathcal{G}, \gamma )$ where $\gamma $ is the composition of $j_{U!}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}* = h_ U^\# $.
Let us construct a functor from $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$. Suppose that $(\mathcal{F}, \varphi )$ is given. For an object $(X, a)$ of $\mathcal{C}/U$ we consider the set $\mathcal{F}_\varphi (X, a)$ of elements $s \in \mathcal{F}(X)$ which under $\varphi $ map to the image of $a \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) = h_ U(X)$ in $h_ U^\# (X)$. It is easy to see that $(X, a) \mapsto \mathcal{F}_\varphi (X, a)$ is a sheaf on $\mathcal{C}/U$. Clearly, the rule $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ defines a functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$.
Consider also the functor $\textit{PSh}(\mathcal{C})/h_ U \to \textit{PSh}(\mathcal{C}/U)$, $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ where $\mathcal{F}_\varphi (X, a)$ is defined as the set of elements of $\mathcal{F}(X)$ mapping to $a \in h_ U(X)$. We claim that the diagram
\[ \xymatrix{ \textit{PSh}(\mathcal{C})/h_ U \ar[r] \ar[d] & \textit{PSh}(\mathcal{C}/U) \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) } \]
commutes, where the vertical arrows are given by sheafification. To see this1, it suffices to prove that the construction commutes with the functor $\mathcal{F} \mapsto \mathcal{F}^+$ of Lemmas 7.10.3 and 7.10.4 and Theorem 7.10.10. Commutation with $\mathcal{F} \mapsto \mathcal{F}^+$ follows from the fact that given $(X, a)$ the categories of coverings of $(X, a)$ in $\mathcal{C}/U$ and coverings of $X$ in $\mathcal{C}$ are canonically identified.
Next, let $\textit{PSh}(\mathcal{C}/U) \to \textit{PSh}(\mathcal{C})/h_ U$ send $\mathcal{G}$ to the pair $(j_{U!}^{PSh}\mathcal{G}, \gamma )$ where $j_{U!}^{PSh}\mathcal{G}$ the presheaf defined by the formula in Lemma 7.25.2 and $\gamma $ is the composition of $j_{U!}^{PSh}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}^{PSh}* = h_ U$ (obvious from the formula). Then it is immediately clear that the diagram
\[ \xymatrix{ \textit{PSh}(\mathcal{C}/U) \ar[r] \ar[d] & \textit{PSh}(\mathcal{C})/h_ U \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# } \]
commutes, where the vertical arrows are sheafification. Putting everything together it suffices to show there are functorial isomorphisms $(j_{U!}^{PSh}\mathcal{G})_\gamma = \mathcal{G}$ for $\mathcal{G}$ in $\textit{PSh}(\mathcal{C}/U)$ and $j_{U!}^{PSh}\mathcal{F}_\varphi = \mathcal{F}$ for $(\mathcal{F}, \varphi )$ in $\textit{PSh}(\mathcal{C})/h_ U$. The value of the presheaf $(j_{U!}^{PSh}\mathcal{G})_\gamma $ on $(X, a)$ is the fibre of the map
\[ \coprod \nolimits _{a' : X \to U} \mathcal{G}(X, a') \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) \]
over $a$ which is $\mathcal{G}(X, a)$. This proves the first equality. The value of the presheaf $j_{U!}^{PSh}\mathcal{F}_\varphi $ is on $X$ is
\[ \coprod \nolimits _{a : X \to U} \mathcal{F}_\varphi (X, a) = \mathcal{F}(X) \]
because given a set map $S \to S'$ the set $S$ is the disjoint union of its fibres.
$\square$
Lemma 7.25.4 says the functor $j_{U!}$ is the composition
\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \rightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \rightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
where the first arrow is an equivalence.
Lemma 7.25.5. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ commutes with fibre products and equalizers (and more generally finite connected limits). In particular, if $\mathcal{F} \subset \mathcal{F}'$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$, then $j_{U!}\mathcal{F} \subset j_{U!}\mathcal{F}'$.
Proof.
Via Lemma 7.25.4 and the fact that an equivalence of categories commutes with all limits, this reduces to the fact that the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \rightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ commutes with fibre products and equalizers. Alternatively, one can prove this directly using the description of $j_{U!}$ in Lemma 7.25.2 using that sheafification is exact. (Also, in case $\mathcal{C}$ has fibre products and equalizers, the result follows from Lemma 7.21.6.)
$\square$
Lemma 7.25.6. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ reflects injections and surjections.
Proof.
We have to show $j_{U!}$ reflects monomorphisms and epimorphisms, see Lemma 7.11.2. Via Lemma 7.25.4 this reduces to the fact that the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ reflects monomorphisms and epimorphisms.
$\square$
Lemma 7.25.7. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. For any sheaf $\mathcal{F}$ on $\mathcal{C}$ we have $j_{U!}j_ U^{-1}\mathcal{F} = \mathcal{F} \times h_ U^\# $.
Proof.
This is clear from the description of $j_{U!}$ in Lemma 7.25.2.
$\square$
Lemma 7.25.8. Let $\mathcal{C}$ be a site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram
\[ \xymatrix{ \mathcal{C}/V \ar[rd]_{j_ V} \ar[rr]_ j & & \mathcal{C}/U \ar[ld]^{j_ U} \\ & \mathcal{C} & } \]
of continuous and cocontinuous functors. The functor $j : \mathcal{C}/V \to \mathcal{C}/U$, $(a : W \to V) \mapsto (f \circ a : W \to U)$ is identified with the functor $j_{V/U} : (\mathcal{C}/U)/(V/U) \to \mathcal{C}/U$ via the identification $(\mathcal{C}/U)/(V/U) = \mathcal{C}/V$. Moreover we have $j_{V!} = j_{U!} \circ j_!$, $j_ V^{-1} = j^{-1} \circ j_ U^{-1}$, and $j_{V*} = j_{U*} \circ j_*$.
Proof.
The commutativity of the diagram is immediate. The agreement of $j$ with $j_{V/U}$ follows from the definitions. By Lemma 7.21.2 we see that the following diagram of morphisms of topoi
7.25.8.1
\begin{equation} \label{sites-equation-relocalize} \vcenter { \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) \ar[rd]_{j_ V} \ar[rr]_ j & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[ld]^{j_ U} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) & } } \end{equation}
is commutative. This proves that $j_ V^{-1} = j^{-1} \circ j_ U^{-1}$ and $j_{V*} = j_{U*} \circ j_*$. The equality $j_{V!} = j_{U!} \circ j_!$ follows formally from adjointness properties.
$\square$
Lemma 7.25.9. Notation $\mathcal{C}$, $f : V \to U$, $j_ U$, $j_ V$, and $j$ as in Lemma 7.25.8. 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 we have
the functor $j^{-1}$ has the following description
\[ j^{-1}(\mathcal{H} \xrightarrow {\varphi } h_ U^\# ) = (\mathcal{H} \times _{\varphi , h_ U^\# , f} h_ V^\# \to h_ V^\# ). \]
the functor $j_!$ has the following description
\[ j_!(\mathcal{H} \xrightarrow {\varphi } h_ V^\# ) = (\mathcal{H} \xrightarrow {h_ f \circ \varphi } h_ U^\# ) \]
Proof.
Proof of (2). Recall that the identification $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ V^\# $ sends $\mathcal{G}$ to $j_{V!}\mathcal{G} \to j_{V!}(*) = h_ V^\# $ and similarly for $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $. Thus $j_!\mathcal{G}$ is mapped to $j_{U!}(j_!\mathcal{G}) \to j_{U!}(*) = h_ U^\# $ and (2) follows because $j_{U!}j_! = j_{V!}$ by Lemma 7.25.8.
The reader can now prove (1) by using that $j^{-1}$ is the right adjoint to $j_!$ and using that the rule in (1) is the right adjoint to the rule in (2). Here is a direct proof. Suppose that $\varphi : \mathcal{H} \to h_ U^\# $ is an object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $. By the proof of Lemma 7.25.4 this corresponds to the sheaf $\mathcal{H}_\varphi $ on $\mathcal{C}/U$ defined by the rule
\[ (a : W \to U) \longmapsto \{ s \in \mathcal{H}(W) \mid \varphi (s) = a\} \]
on $\mathcal{C}/U$. The pullback $j^{-1}\mathcal{H}_\varphi $ to $\mathcal{C}/V$ is given by the rule
\[ (a : W \to V) \longmapsto \{ s \in \mathcal{H}(W) \mid \varphi (s) = f \circ a\} \]
by the description of $j^{-1} = j_{U/V}^{-1}$ as the restriction of $\mathcal{H}_\varphi $ to $\mathcal{C}/V$. On the other hand, applying the rule to the object
\[ \xymatrix{ \mathcal{H}' = \mathcal{H} \times _{\varphi , h_ U^\# , f} h_ V^\# \ar[rr]^-{\varphi '} & & h_ V^\# } \]
of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ V^\# $ we get $\mathcal{H}'_{\varphi '}$ given by
\begin{align*} (a : W \to V) \longmapsto & \{ s' \in \mathcal{H}'(W) \mid \varphi '(s') = a\} \\ = & \{ (s, a') \in \mathcal{H}(W) \times h_ V^\# (W) \mid a' = a \text{ and } \varphi (s) = f \circ a'\} \end{align*}
which is exactly the same rule as the one describing $j^{-1}\mathcal{H}_\varphi $ above.
$\square$
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