## 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

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

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

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

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. $\square$

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