The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.43 Subtopoi

Here is the definition.

Definition 7.43.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is called an embedding if $f_*$ is fully faithful.

According to Lemma 7.41.1 this is equivalent to asking the adjunction map $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ to be an isomorphism for every sheaf $\mathcal{F}$ on $\mathcal{D}$.

Definition 7.43.2. Let $\mathcal{C}$ be a site. A strictly full subcategory $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is a subtopos if there exists an embedding of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ such that $E$ is equal to the essential image of the functor $f_*$.

The subtopoi constructed in the following lemma will be dubbed "open" in the definition later on.

Lemma 7.43.3. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. The following are equivalent

  1. $\mathcal{F}$ is a subobject of the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

  2. the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is a subtopos of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Proof. We have seen in Lemma 7.30.1 that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is a topos. In fact, we recall the proof. First we apply Lemma 7.29.5 to see that we may assume $\mathcal{C}$ is a site with a subcanonical topology, fibre products, a final object $X$, and an object $U$ with $\mathcal{F} = h_ U$. The proof of Lemma 7.30.1 shows that the morphism of topoi $j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is equal (modulo certain identifications) to the localization morphism $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Assume (2). This means that $j_ U^{-1}j_{U, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism for all sheaves $\mathcal{G}$ on $\mathcal{C}/U$. For any object $Z/U$ of $\mathcal{C}/U$ we have

\[ (j_{U, *}h_{Z/U})(U) = \mathop{Mor}\nolimits _{\mathcal{C}/U}(U \times _ X U/U, Z/U) \]

by Lemma 7.27.2. Setting $\mathcal{G} = h_{Z/U}$ in the equality above we obtain

\[ \mathop{Mor}\nolimits _{\mathcal{C}/U}(U \times _ X U/U, Z/U) = \mathop{Mor}\nolimits _{\mathcal{C}/U}(U, Z/U) \]

for all $Z/U$. By Yoneda's lemma (Categories, Lemma 4.3.5) this implies $U \times _ X U = U$. By Categories, Lemma 4.13.3 $U \to X$ is a monomorphism, in other words (1) holds.

Assume (1). Then $j_ U^{-1} j_{U, *} = \text{id}$ by Lemma 7.27.4. $\square$

Definition 7.43.4. Let $\mathcal{C}$ be a site. A strictly full subcategory $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is an open subtopos if there exists a subsheaf $\mathcal{F}$ of the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ such that $E$ is the subtopos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ described in Lemma 7.43.3.

This means there is a bijection between the collection of open subtopoi of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and the set of subobjects of the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Given an open subtopos there is a "closed" complement.

Lemma 7.43.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a subsheaf of the final object $*$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. The full subcategory of sheaves $\mathcal{G}$ such that $\mathcal{F} \times \mathcal{G} \to \mathcal{F}$ is an isomorphism is a subtopos of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Proof. We apply Lemma 7.29.5 to see that we may assume $\mathcal{C}$ is a site with the properties listed in that lemma. In particular $\mathcal{C}$ has a final object $X$ (so that $* = h_ X$) and an object $U$ with $\mathcal{F} = h_ U$.

Let $\mathcal{D} = \mathcal{C}$ as a category but a covering is a family $\{ V_ j \to V\} $ of morphisms such that $\{ V_ i \to V\} \cup \{ U \times _ X V \to V\} $ is a covering. By our choice of $\mathcal{C}$ this means exactly that

\[ h_{U \times _ X V} \amalg \coprod h_{V_ i} \longrightarrow h_ V \]

is surjective. We claim that $\mathcal{D}$ is a site, i.e., the coverings satisfy the conditions (1), (2), (3) of Definition 7.6.2. Condition (1) holds. For condition (2) suppose that $\{ V_ i \to V\} $ and $\{ V_{ij} \to V_ i\} $ are coverings of $\mathcal{D}$. Then the composition

\[ \coprod \left( h_{U \times _ X V_ i} \amalg \coprod h_{V_{ij}} \right) \longrightarrow h_{U \times _ X V} \amalg \coprod h_{V_ i} \longrightarrow h_ V \]

is surjective. Since each of the morphisms $U \times _ X V_ i \to V$ factors through $U \times _ X V$ we see that

\[ h_{U \times _ X V} \amalg \coprod h_{V_{ij}} \longrightarrow h_ V \]

is surjective, i.e., $\{ V_{ij} \to V\} $ is a covering of $V$ in $\mathcal{D}$. Condition (3) follows similarly as a base change of a surjective map of sheaves is surjective.

Note that the (identity) functor $u : \mathcal{C} \to \mathcal{D}$ is continuous and commutes with fibre products and final objects. Hence we obtain a morphism $f : \mathcal{D} \to \mathcal{C}$ of sites (Proposition 7.14.7). Observe that $f_*$ is the identity functor on underlying presheaves, hence fully faithful. To finish the proof we have to show that the essential image of $f_*$ is the full subcategory $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ singled out in the lemma. To do this, note that $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}))$ is in $E$ if and only if $\mathcal{G}(U \times _ X V)$ is a singleton for all objects $V$ of $\mathcal{C}$. Thus such a sheaf satisfies the sheaf property for all coverings of $\mathcal{D}$ (argument omitted). Conversely, if $\mathcal{G}$ satisfies the sheaf property for all coverings of $\mathcal{D}$, then $\mathcal{G}(U \times _ X V)$ is a singleton, as in $\mathcal{D}$ the object $U \times _ X V$ is covered by the empty covering. $\square$

Definition 7.43.6. Let $\mathcal{C}$ be a site. A strictly full subcategory $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is an closed subtopos if there exists a subsheaf $\mathcal{F}$ of the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ such that $E$ is the subtopos described in Lemma 7.43.5.

All right, and now we can define what it means to have a closed immersion and an open immersion of topoi.

Definition 7.43.7. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi.

  1. We say $f$ is an open immersion if $f$ is an embedding and the essential image of $f_*$ is an open subtopos.

  2. We say $f$ is a closed immersion if $f$ is an embedding and the essential image of $f_*$ is a closed subtopos.

Lemma 7.43.8. Let $i : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a closed immersion of topoi. Then $i_*$ is fully faithful, transforms surjections into surjections, commutes with coequalizers, commutes with pushouts, reflects injections, reflects surjections, and reflects bijections.

Proof. Let $\mathcal{F}$ be a subsheaf of the final object $*$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and let $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be the full subcategory consisting of those $\mathcal{G}$ such that $\mathcal{F} \times \mathcal{G} \to \mathcal{F}$ is an isomorphism. By Lemma 7.43.5 the functor $i_*$ is isomorphic to the inclusion functor $\iota : E \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Let $j_{\mathcal{F}} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be the localization functor (Lemma 7.30.1). Note that $E$ can also be described as the collection of sheaves $\mathcal{G}$ such that $j_\mathcal {F}^{-1}\mathcal{G} = *$.

Let $a, b : \mathcal{G}_1 \to \mathcal{G}_2$ be two morphism of $E$. To prove $\iota $ commutes with coequalizers it suffices to show that the coequalizer of $a$, $b$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ lies in $E$. This is clear because the coequalizer of two morphisms $* \to *$ is $*$ and because $j_\mathcal {F}^{-1}$ is exact. Similarly for pushouts.

Thus $i_*$ satisfies properties (5), (6), and (7) of Lemma 7.41.1 and hence the morphism $i$ satisfies all properties mentioned in that lemma, in particular the ones mentioned in this lemma. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08LT. Beware of the difference between the letter 'O' and the digit '0'.