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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

  1. the functor j_\mathcal {F}^{-1} is the functor \mathcal{H} \mapsto \mathcal{H} \times \mathcal{F}/\mathcal{F}, and

  2. 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}.

  1. 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}.

  2. 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{G}) = (\mathcal{E} \xrightarrow {s \circ e} \mathcal{F}).

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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