## 7.31 Localization and morphisms of topoi

This section is the analogue of Section 7.28 for morphisms of topoi.

Lemma 7.31.1. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then there exists a commutative diagram of topoi

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[r]_{j_\mathcal {F}} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G} \ar[r]^{j_\mathcal {G}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). }$

The morphism $f'$ is characterized by the property that

$(f')^{-1}(\mathcal{H} \xrightarrow {\varphi } \mathcal{G}) = (f^{-1}\mathcal{H} \xrightarrow {f^{-1}\varphi } \mathcal{F})$

and we have $f'_*j_\mathcal {F}^{-1} = j_\mathcal {G}^{-1}f_*$.

Proof. Since the statement is about topoi and does not refer to the underlying sites we may change sites at will. Hence by the discussion in Remark 7.29.7 we may assume that $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ satisfying the assumptions of Proposition 7.14.7 between sites having all finite limits and subcanonical topologies, and such that $\mathcal{G} = h_ V$ for some object $V$ of $\mathcal{D}$. Then $\mathcal{F} = f^{-1}h_ V = h_{u(V)}$ by Lemma 7.13.5. By Lemma 7.28.1 we obtain a commutative diagram of morphisms of topoi

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}), }$

and we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$. By Lemma 7.30.5 we may identify $j_\mathcal {F}$ and $j_ U$ and $j_\mathcal {G}$ and $j_ V$. The description of $(f')^{-1}$ is given in Lemma 7.28.1. $\square$

Lemma 7.31.2. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$. Set $U = u(V)$. Set $\mathcal{G} = h_ V^\#$, and $\mathcal{F} = h_ U^\# = f^{-1}h_ V^\#$ (see Lemma 7.13.5). Then the diagram of morphisms of topoi of Lemma 7.31.1 agrees with the diagram of morphisms of topoi of Lemma 7.28.1 via the identifications $j_\mathcal {F}= j_ U$ and $j_\mathcal {G} = j_ V$ of Lemma 7.30.5.

Proof. This is not a complete triviality as the choice of morphism of sites giving rise to $f$ made in the proof of Lemma 7.31.1 may be different from the morphisms of sites given to us in the lemma. But in both cases the functor $(f')^{-1}$ is described by the same rule. Hence they agree and the associated morphism of topoi is the same. Some details omitted. $\square$

Lemma 7.31.3. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $s : \mathcal{F} \to f^{-1}\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of topoi

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[r]_{j_\mathcal {F}} \ar[d]_{f_ s} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G} \ar[r]^{j_\mathcal {G}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). }$

We have $f_ s = f' \circ j_{\mathcal{F}/f^{-1}\mathcal{G}}$ where $f' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{F}$ is as in Lemma 7.31.1 and $j_{\mathcal{F}/f^{-1}\mathcal{G}} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G}$ is as in Lemma 7.30.6. The functor $(f_ s)^{-1}$ is described by the rule

$(f_ s)^{-1}(\mathcal{H} \xrightarrow {\varphi } \mathcal{G}) = (f^{-1}\mathcal{H} \times _{f^{-1}\varphi , f^{-1}\mathcal{G}, s} \mathcal{F} \rightarrow \mathcal{F}).$

Finally, given any morphisms $b : \mathcal{G}' \to \mathcal{G}$, $a : \mathcal{F}' \to \mathcal{F}$ and $s' : \mathcal{F}' \to f^{-1}\mathcal{G}'$ such that

$\xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_ a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} }$

commutes, then the diagram

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}' \ar[r]_{j_{\mathcal{F}'/\mathcal{F}}} \ar[d]_{f_{s'}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[d]^{f_ s} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}' \ar[r]^{j_{\mathcal{G}'/\mathcal{G}}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}. }$

commutes.

Proof. The commutativity of the first square follows from the commutativity of the diagram in Lemma 7.30.6 and the commutativity of the diagram in Lemma 7.31.1. The description of $f_ s^{-1}$ follows on combining the descriptions of $(f')^{-1}$ in Lemma 7.31.1 with the description of $(j_{\mathcal{F}/f^{-1}\mathcal{G}})^{-1}$ in Lemma 7.30.6. The commutativity of the last square then follows from the equality

$f^{-1}\mathcal{H} \times _{f^{-1}\mathcal{G}, s} \mathcal{F} \times _\mathcal {F} \mathcal{F}' = f^{-1}(\mathcal{H} \times _\mathcal {G} \mathcal{G}') \times _{f^{-1}\mathcal{G}', s'} \mathcal{F}'$

which is formal. $\square$

Lemma 7.31.4. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$. Let $c : U \to u(V)$ be a morphism. Set $\mathcal{G} = h_ V^\#$ and $\mathcal{F} = h_ U^\# = f^{-1}h_ V^\#$. Let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be the map induced by $c$. Then the diagram of morphisms of topoi of Lemma 7.28.3 agrees with the diagram of morphisms of topoi of Lemma 7.31.3 via the identifications $j_\mathcal {F} = j_ U$ and $j_\mathcal {G} = j_ V$ of Lemma 7.30.5.

Proof. This follows on combining Lemmas 7.30.7 and 7.31.2. $\square$

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