nLab
geometric embedding

Idea

A geometric embedding is the right notion of embedding or inclusion of topoi $F\hookrightarrow E$.

Notably the inclusion $\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S)$ of a category of sheaves into its presheaf topos or more generally the inclusion ${\mathrm{Sh}}_{j}E\hookrightarrow E$ of sheaves in a topos $E$ into $E$ itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi.

Another perspective is that a geometric embedding $F\hookrightarrow E$ is the localizations of $E$ at the class $W$ or morphisms that the left adjoint $E\to F$ sends to isomorphisms in $F$.

Definition

For $F$ and $E$ two topoi, a geometric morphism

$$F\stackrel{f}{\to }EF\stackrel{\stackrel{f_{*}}{\to }}{\stackrel{f^{*}}{\leftarrow }}E$$F \stackrel{f}{\to} E \;\;\;\; F \stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}} E

is a geometric embedding if the following equivalent conditions are satisfied

• the direct image functor $f_{*}$ is full and faithful (so that $F$ is a full subcategory of $E$);

• the counit $ϵ:f^{*}{f}_{*}\to {\mathrm{Id}}_F$ of the adjunction $(f^{*}⊣f_{*})$ is an isomorphism

• there is a Lawvere-Tierney topology on $E$ and an equivalence of categories $e:F\stackrel{\simeq }{\to }{\mathrm{Sh}}_{j}E$ such that the diagram of geometric morphisms $\begin{array}{ccc}F& \stackrel{f_{*}}{\to }& E\\ & {}_e{\searrow }^{\simeq }& {\uparrow }^i\\ & & {\mathrm{Sh}}_{j}E\end{array}$ commutes up to natural isomorphism $e^{*}{i}^{*}\simeq f^{*}$

That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of

• MacLane-Moerdijk, Sheaves in geometry and logic

Relation to localization

There is a close relation between geometric embedding and localization.

Let $f:F\hookrightarrow E$ be a geometric embedding and let $W\subset \mathrm{Mor}(E)$ be the class of morphisms sent by $f^{*}$ to isomorphisms in $F$.

Theorem

We have:

• $F$ is equivalent to the localization $E[W{]}^{-1}$;
• $F$ is equivalent to the full subcategory of $E$ on $W$-local objects.

This fact connects for instance the description of sheafification in terms of geometric embedding $\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S)$ as described for instance in

• MacLane-Moerdijk, Sheaves in Geometry and Logic

with that in terms of localization at local isomorphisms, as described in

• Kashiwara-Schapira, Categories and Sheaves.

Moreover, this is the basis on which sheafification is generalized to (∞,1)-sheafification in

• Jacob Lurie, Higher Topos Theory .

The following gives a detailed proof of the above assertion.

Write $\eta :{\mathrm{Id}}_E\to f_{*}{f}^{*}$ for the unit of the adjunction.

Since $f_{*}$ is fully faithful we will identify objects and morphism of $F$ with their images in $E$. To further trim down the notation write $\overline{(-)}:=f^{*}$ for the left adjoint.

Definition

Write $W$ for the class of morphism that are sent to isomorphism under $f^{*}$,

$$W=(f^{*}{)}^{-1}\{g:c\stackrel{\simeq }{\to }d\in \mathrm{Mor}(E)\}.$$W = (f^*)^{-1}\{g: c\stackrel{\simeq}{\to} d \in Mor(E)\} \,.

Proposition

$E$ equipped with the class $W$ is a category with weak equivalences, in that $W$ satisfies 2-out-of-3.

Proof

Follows since isomorphisms satisfy 2-out-of-3.

Proposition

$W$ is a left multiplicative system.

Proof

This follows using the fact that $f^{*}$ is left exact and hence preserves finite limits.

In more detail:

We have already seen in the previous proposition that

• every isomorphism is in $W$;

• $W$ is closed under composition.

It remains to check the following points:

Given any

$$\begin{array}{ccc}& & a\\ & & {\downarrow }^w\\ b& \stackrel{h}{\to }& c\end{array}$$\array{ && a \\ && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with $w\in W$, we have to show that there is

$$\begin{array}{ccc}d& \to & a\\ {\downarrow }^{w\prime }& & {\downarrow }^w\\ b& \stackrel{h}{\to }& c\end{array}$$\array{ d &\to& a \\ \downarrow^{w'} && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with $w\prime \in W$.

To get this, take this to be the pullback diagram, $w\prime :=h^{*}w$. Since $f^{*}$ preserves pullbacks, it follows that

$$\begin{array}{ccc}\overline{d}& \to & \overline{a}\\ {\downarrow }^{\overline{w}\prime }& & {\downarrow }^{\overline{w}}\\ \overline{b}& \stackrel{\overline{h}}{\to }& \overline{c}\end{array}$$\array{ \bar d &\to& \bar a \\ \downarrow^{\bar w'} && \downarrow^{\bar w} \\ \bar b &\stackrel{\bar h}{\to}& \bar c }

is a pullback diagram in $F$ with $\overline{w}\prime ={\overline{h}}^{*}\overline{w}$. But by assumption $\overline{w}$ is an isomorphism. Therefore $\overline{w}\prime$ is an isomorphism, therefore $w\prime$ is in $W$.

Finally for every

$$a\stackrel{\stackrel{r}{\to }}{\stackrel{s}{\to }}b\stackrel{w}{\to }c$$a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b \stackrel{w}{\to} c

with $w\in W$ such that the two composites coincide, we need to find

$$d\stackrel{w\prime }{\to }a\stackrel{\stackrel{r}{\to }}{\stackrel{s}{\to }}b$$d \stackrel{w'}{\to} a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b

with $w\prime \in W$ such that the composites again coincide.

To get this, take $w\prime$ to be the equalizer of the two morphisms. Sending everything with $f^{*}$ to $F$ we find from

$$\overline{a}\stackrel{\stackrel{\overline{r}}{\to }}{\stackrel{\overline{s}}{\to }}b\stackrel{\overline{w}}{\to }c$$\bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c

that $\overline{r}=\overline{s}$, since $\overline{w}$ is an isomorphism. This implies that $\overline{w}\prime$ is the equalizer

$$\overline{d}\stackrel{\overline{w}\prime }{\to }a\stackrel{\stackrel{\overline{r}}{\to }}{\stackrel{\overline{s}}{\to }}b$$\bar d \stackrel{\bar w'}{\to} a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b

of two equal morphism, hence an identity. So $w\prime$ is in $W$.

Proposition

For every object $a\in E$

• the unit ${\eta }_a:a\to \overline{a}$ is in $W$;

• if $a$ is already in $F$ then the unit is already an isomorphism.

Proof

This follows from the zig-zag identities of the adjoint functors.

$$\begin{array}{cccc}& \nearrow & {\Downarrow }^{\eta }& {\searrow }^{{\mathrm{Id}}_E}\\ E& \stackrel{\overline{(-)}}{\to }& F& \hookrightarrow & E& \stackrel{\overline{(-)}}{\to }& F\\ & & & \searrow & {\Downarrow }^{\simeq }& {\nearrow }_{{\mathrm{Id}}_F}\end{array}=\begin{array}{cc}& \nearrow {\searrow }^{\overline{(-)}}\\ E& {\Downarrow }^{\mathrm{Id}}& F\\ & \searrow {\nearrow }_{\overline{(-)}}\end{array}$$\array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F \\ &&& \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow^{\bar{(-)}} \\ E &\Downarrow^{Id}& F \\ & \searrow \nearrow_{\bar{(-)}} }

and

$$\begin{array}{cccccc}& & & \nearrow & {\Downarrow }^{\eta }& {\searrow }^{{\mathrm{Id}}_E}\\ F& \hookrightarrow & E& \stackrel{\overline{(-)}}{\to }& F& \hookrightarrow & E\\ & \searrow & {\Downarrow }^{\simeq }& {\nearrow }_{{\mathrm{Id}}_F}\end{array}=\begin{array}{cc}& \nearrow \searrow \\ F& {\Downarrow }^{\mathrm{Id}}& E\\ & \searrow \nearrow \end{array}$$\array{ &&& \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow \\ F &\Downarrow^{Id}& E \\ & \searrow \nearrow }

In components they say that

• for every $a\in E$ we have $(\overline{a}\stackrel{{\overline{\eta }}_a}{\to }\overline{\overline{a}}\stackrel{\simeq }{\to }\overline{a})={\mathrm{Id}}_{\overline{a}}$

• for every $a\in F$ we have $(a\stackrel{{\eta }_a}{\to }\overline{a}\stackrel{\simeq }{\to }a)={\mathrm{Id}}_a$

This implies the claim.

Definition

An object $a\in E$ is $W$-local object if for every $g:c\to d$ in $W$ the map

$$g^{*}:{\mathrm{Hom}}_E(d,a)\stackrel{\simeq }{\to }{\mathrm{Hom}}_E(c,a)$$g^* : Hom_E(d,a) \stackrel{\simeq}{\to} Hom_E(c,a)

obtained by precomposition is an isomorphism.

Proposition

Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$

Proof

First assume that $a\in F$. We need to show that $a$ is $W$-local.

Notice that the existence of the required isomorphism ${\mathrm{Hom}}_F(d,a)\simeq {\mathrm{Hom}}_F(c,a)$ is equivalent to the statement that for every diagram

$$\begin{array}{ccc}c& \stackrel{}{\to }& d\\ {\downarrow }^h\\ a\end{array}$$\array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} \\ a }

there is a unique extension

$$\begin{array}{ccc}c& \stackrel{}{\to }& d\\ {\downarrow }^h& \swarrow \\ a\end{array}.$$\array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} & \swarrow \\ a } \,.

To see the existence of this extension, hit the original diagram with $f^{*}$ to get

$$\begin{array}{ccc}\overline{c}& \stackrel{\simeq }{\to }& \overline{d}\\ {\downarrow }^{\overline{h}}\\ \overline{a}\simeq a\end{array}.$$\array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} \\ \bar a \simeq a } \,.

By the assumption that $c\to d$ is in $W$ the morphism $\overline{c}\to \overline{d}$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\overline{a}\simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

$$\begin{array}{ccc}\overline{c}& \stackrel{\simeq }{\to }& \overline{d}\\ {\downarrow }^{\overline{h}}& {\swarrow }_{\exists !k}\\ \overline{a}\simeq a\end{array}.$$\array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} & \swarrow_{\exists ! k} \\ \bar a \simeq a } \,.

By the hom-isomorphism (using full faithfullness of $f_{*}$ to work entirely in $E$)

$${\mathrm{Hom}}_E(\overline{d},a)\simeq {\mathrm{Hom}}_E(d,a)$$Hom_E(\bar d, a) \simeq Hom_E(d,a)

this defines a morphism $k:d\to a$. Chasing $k$ through the naturality diagram of the hom-isomorphism

$$\begin{array}{ccc}{\mathrm{Hom}}_E(\overline{d},\overline{a})& \stackrel{\simeq }{\to }& {\mathrm{Hom}}_E(d,\overline{a})\\ \downarrow & & \downarrow \\ {\mathrm{Hom}}_E(\overline{c},\overline{a})& \stackrel{\simeq }{\to }& {\mathrm{Hom}}_E(c,\overline{a})\end{array}.$$\array{ Hom_E(\bar d, \bar a) &\stackrel{\simeq}{\to}& Hom_E(d,\bar a) \\ \downarrow && \downarrow \\ Hom_E(\bar c, \bar a) &\stackrel{\simeq}{\to}& Hom_E(c,\bar a) } \,.

shows that $k:d\to a$ does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So $a\in F$ is $W$-local.

Now for the converse, assume that a given $a$ is $W$-local.

By one of the above propositions we know that the unit ${\eta }_a:a\to \overline{a}$ is in $W$, so by the $W$-locality of $a$ it follows that

$$\begin{array}{ccc}a& \stackrel{{\eta }_a}{\to }& \overline{a}\\ {\downarrow }^{{\mathrm{Id}}_a}\\ a\end{array}$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a }

has an extension

$$\begin{array}{ccc}a& \stackrel{{\eta }_a}{\to }& \overline{a}\\ {\downarrow }^{{\mathrm{Id}}_a}& {\swarrow }_{{\rho }_a}\\ a\end{array}.$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} & \swarrow_{\rho_a} \\ a } \,.

By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that ${\mathrm{Id}}_a$, being an isomorphism, is in $W$) it follows that ${\rho }_a:\overline{a}\to a$ is in $W$.

Since $\overline{a}$ is in $F$ and therefore $W$-local by the above, it follows that also

$$\begin{array}{ccc}\overline{a}& \stackrel{{\rho }_a}{\to }& a\\ {\downarrow }^{{\mathrm{Id}}_{\overline{a}}}\\ \overline{a}\end{array}$$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} \\ \bar a }

has an extension

$$\begin{array}{ccc}\overline{a}& \stackrel{{\rho }_a}{\to }& a\\ {\downarrow }^{{\mathrm{Id}}_{\overline{a}}}& {\swarrow }_{{\lambda }_a}\\ \overline{a}\end{array}.$$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a} \\ \bar a } \,.

So ${\eta }_a$ has a left inverse ${\rho }_a$ which itself has a left inverse ${\lambda }_a$. It follows that ${\rho }_a$ is also a right inverse to ${\eta }_a$, since

$$\begin{array}{rl}\stackrel{{\rho }_a}{\to }\stackrel{{\eta }_a}{\to }& =\stackrel{{\rho }_a}{\to }\stackrel{{\eta }_a}{\to }{\underbrace{\stackrel{{\rho }_a}{\to }\stackrel{{\lambda }_a}{\to }}}_{\mathrm{Id}}\\ & =\stackrel{{\rho }_a}{\to }{\underbrace{\stackrel{{\eta }_a}{\to }\stackrel{{\rho }_a}{\to }}}_{\mathrm{Id}}\stackrel{{\lambda }_a}{\to }\\ & =\stackrel{{\rho }_a}{\to }\stackrel{{\lambda }_a}{\to }\\ & =\mathrm{Id}\end{array}.$$\begin{aligned} \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} & = \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} \underbrace{ \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} }_{Id} \\ & = \stackrel{\rho_a}{\to} \underbrace{ \stackrel{\eta_a}{\to} \stackrel{\rho_a}{\to} }_{Id} \stackrel{\lambda_a}{\to} \\ &= \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} \\ &= Id \end{aligned} \,.

So if $a$ is $W$-local we find that ${\eta }_a:a\to \overline{a}$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.

Corollary

$F$ is equivalent to the full subcategory $E_{W-\mathrm{loc}}$ of $E$ on $W$-local objects.

Proof

By standard reasoning (e.g. KS lemma 1.3.11) there is a functor $F\to E_{W-\mathrm{loc}}$ and a natural isomorphism

$$\begin{array}{ccccc}F& & \hookrightarrow & & E\\ & \searrow & {\Downarrow }^{\simeq }& \nearrow \\ & & E_{W-\mathrm{loc}}\end{array}.$$\array{ F &&\hookrightarrow&& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow \\ && E_{W-loc} } \,.

Since $F\hookrightarrow E$ and $E_{W-\mathrm{loc}}\hookrightarrow E$ are full and faithful, so is $F\to E_{W-\mathrm{loc}}$. Since by the above it is also essentially surjective, it establishes the equivalence $F\simeq E_{W-\mathrm{loc}}$.

Proposition

$F$ is equivalent to the localization $E[W^{-1}]$ of $E$ at $W$.

Proof

By one of the above propositions we know that $W$ is a left multiplicative systems.

This implies that the localization $E[W^{-1}]$ is (equivalent to) the category with the same objects as $E$, and with hom-sets given by

$${\mathrm{Hom}}_{E[W^{-1}]}(a,b)=\underset{a\prime \stackrel{p\in W}{\to }a}{\mathrm{colim}}{\mathrm{Hom}}_E(a\prime ,b).$$Hom_{E[W^{-1}]}(a,b) = \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_E(a',b) \,.

There is an obvious candidate for a functor

$$F\to E[W^{-1}]$$F \to E[W^{-1}]

given on objects by the usual embedding by $f_{*}$ and on morphism by the map which regards a morphism trivially as a span with left leg the identity

$$(a\to b)\mapsto \left(\begin{array}{ccc}a& \to & b\\ {\downarrow }^{{\mathrm{Id}}_a}\\ a\end{array}\right).$$(a \to b) \;\; \mapsto \;\; \left( \array{ a &\to& b \\ \downarrow^{Id_a} \\ a } \right) \,.

For this to be an equivalence of categories we need to show that this is a essentially surjective and full and faithful functor.

To see essential surjectivity, let $a$ be any object in $E$ and let ${\eta }_a:a\to \overline{a}$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, ${\eta }_a$ is in $W$. This means that the span

$$\begin{array}{ccc}a& \stackrel{{\mathrm{Id}}_a}{\to }& a\\ {\downarrow }^{{\eta }_a}\\ \overline{a}\end{array}$$\array{ a &\stackrel{Id_a}{\to}& a \\ \downarrow^{\eta_a} \\ \bar a }

represents an element in ${\mathrm{Hom}}_{E[W^{-1}]}(\overline{a},a)$, and this element is clearly an isomorphism: the inverse is represented by

$$\begin{array}{ccc}a& \stackrel{{\eta }_a}{\to }& \overline{a}\\ {\downarrow }^{{\mathrm{Id}}_a}\\ a\end{array}.$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a } \,.

Since every $\overline{a}$ is in the image of our functor, this shows that it is essentially surjective.

To see fullness and faithfulness, let $a,b\in F$ be any two objects. By one of the above propositions this means in particular that $b$ is a $W$-local object. As discussed above, this means that every span

$$\begin{array}{ccc}a\prime & \to & b\\ {\downarrow }^w\\ a\end{array}$$\array{ a' &\to& b \\ \downarrow^w \\ a }

with $w\in W$ has a unique extension

$$\begin{array}{ccc}a\prime & \to & b\\ {\downarrow }^w& \nearrow \\ a\end{array}.$$\array{ a' &\to& b \\ \downarrow^w & \nearrow \\ a } \,.

But this implies that in the colimit that defines the hom-set of $E[W^{-1}]$ all these spans are identified with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in ${\mathrm{Hom}}_E(a,b)\simeq {\mathrm{Hom}}_F(a,b)$ so that indeed

$${\mathrm{Hom}}_{E[W^{-1}]}(a,b)\simeq {\mathrm{Hom}}_F(a,b)$$Hom_{E[W^{-1}]}(a,b) \simeq Hom_{F}(a,b)

for all $a,b\in F$. Hence our functor is also full and faithful and therefore define an equivalence of categories

$$F\stackrel{\simeq }{\to }E[W^{-1}].$$F \stackrel{\simeq}{\to} E[W^{-1}] \,.

References

section VII, 4 of

• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic