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geometric embedding

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Topos Theory

topos theory

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Extra stuff, structure, properties

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In higher category theory

Theorems

Contents

Idea

A geometric embedding is the right notion of embedding or inclusion of topoi FEF \hookrightarrow E, i.e. of subtoposes.

Notably the inclusion Sh(S)PSh(S)Sh(S) \hookrightarrow PSh(S) of a category of sheaves into its presheaf topos or more generally the inclusion Sh jEESh_j E \hookrightarrow E of sheaves in a topos EE into EE 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 FEF \hookrightarrow E is the localizations of EE at the class WW or morphisms that the left adjoint EFE \to F sends to isomorphisms in FF.

The induced geometric morphism of a topological immersion XYX \hookrightarrow Y is a geometric embedding. The converse holds if YY is a T 0T_0 space. (Example A4.2.12(c) in (Johnstone))

Definition

For FF and EE two topoi, a geometric morphism

FfEFf *f *E F \stackrel{f}{\to} E \;\;\;\; F \stackrel{\stackrel{f_*}{\to}}{\underset{f^*}{\leftarrow}} E

is a geometric embedding if the following equivalent conditions are satisfied

  • the direct image functor f *f_* is full and faithful (so that FF is a full subcategory of EE);

  • the counit ϵ:f *f *Id F\epsilon : f^* f_* \to Id_{F} of the adjunction (f *f *)(f^* \dashv f_*) is an isomorphism

  • there is a Lawvere-Tierney topology on EE and an equivalence of categories e:FSh jEe : F \stackrel{\simeq}{\to} Sh_j E such that the diagram of geometric morphisms F f * E e i Sh jE\array{ F &\stackrel{f_*}{\to}& E \\ & {}_{e}\searrow^\simeq & \uparrow^{i} \\ && Sh_j E} commutes up to natural isomorphism e *i *f *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 (MacLaneMoerdijk)

Properties

Relation to localization

There is a close relation between geometric embedding and localization.

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

Theorem

We have:

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

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

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

The following gives a detailed proof of the above assertion.

Write η:Id Ef *f *\eta : Id_E \to f_* f^* for the unit of the adjunction.

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

Definition

Write WW for the class of morphism that are sent to isomorphism under f *f^*,

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

EE equipped with the class WW is a category with weak equivalences, in that WW satisfies 2-out-of-3.

Proof

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

Proposition

WW is a left multiplicative system.

Proof

This follows using the fact that f *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 WW;

  • WW is closed under composition.

It remains to check the following points:

Given any

a w b h c \array{ && a \\ && \downarrow^w \\ b &\stackrel{h}{\to}& c }

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

d a w w b h c \array{ d &\to& a \\ \downarrow^{w'} && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with wWw' \in W.

To get this, take this to be the pullback diagram, w:=h *ww' := h^* w. Since f *f^* preserves pullbacks, it follows that

d¯ a¯ w¯ w¯ b¯ h¯ c¯ \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 FF with w¯=h¯ *w¯\bar w' = \bar h^* \bar w. But by assumption w¯\bar w is an isomorphism. Therefore w¯\bar w' is an isomorphism, therefore ww' is in WW.

Finally for every

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

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

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

with wWw' \in W such that the composites again coincide.

To get this, take ww' to be the equalizer of the two morphisms. Sending everything with f *f^* to FF we find from

a¯s¯r¯bw¯c \bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c

that r¯=s¯\bar r = \bar s, since w¯\bar w is an isomorphism. This implies that w¯\bar w' is the equalizer

d¯w¯as¯r¯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 ww' is in WW.

Proposition

For every object aEa \in E

  • the unit η a:aa¯\eta_a : a \to \bar a is in WW;

  • if aa is already in FF then the unit is already an isomorphism.

Proof

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

η Id E E ()¯ F E ()¯ F Id F= ()¯ E Id F ()¯ \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

η Id E F E ()¯ F E Id F= F Id E \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 aEa \in E we have (a¯η¯ aa¯¯a¯)=Id a¯(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}

  • for every aFa \in F we have (aη aa¯a)=Id a(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a) = Id_a

This implies the claim.

Definition

An object aEa \in E is WW-local object if for every g:cdg : c \to d in WW the map

g *:Hom E(d,a)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 WW-local objects are precisely the objects of FF in EE

Proof

First assume that aFa \in F. We need to show that aa is WW-local.

Notice that the existence of the required isomorphism Hom F(d,a)Hom F(c,a) Hom_F(d,a) \simeq Hom_F(c,a) is equivalent to the statement that for every diagram

c d h a \array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} \\ a }

there is a unique extension

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

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

c¯ d¯ h¯ a¯a. \array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} \\ \bar a \simeq a } \,.

By the assumption that cdc \to d is in WW the morphism c¯d¯\bar c \to \bar d here is an isomorphism. By the assumption that aa is already in FF we have a¯a\bar a \simeq a since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

c¯ d¯ h¯ !k a¯a. \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 *f_* to work entirely in EE)

Hom E(d¯,a)Hom E(d,a) Hom_E(\bar d, a) \simeq Hom_E(d,a)

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

Hom E(d¯,a¯) Hom E(d,a¯) Hom E(c¯,a¯) Hom E(c,a¯). \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:dak : d \to a does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So aFa \in F is WW-local.

Now for the converse, assume that a given aa is WW-local.

By one of the above propositions we know that the unit η a:aa¯\eta_a : a \to \bar a is in WW, so by the WW-locality of aa it follows that

a η a a¯ Id a a \array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a }

has an extension

a η a a¯ Id a ρ a a. \array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} & \swarrow_{\rho_a} \\ a } \,.

By the 2-out-of-3 property of WW shown in one of the above propositions, (using that Id aId_a, being an isomorphism, is in WW) it follows that ρ a:a¯a\rho_a : \bar a \to a is in WW.

Since a¯\bar a is in FF and therefore WW-local by the above, it follows that also

a¯ ρ a a Id a¯ a¯ \array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} \\ \bar a }

has an extension

a¯ ρ a a Id a¯ λ a a¯. \array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a} \\ \bar a } \,.

So η a\eta_a has a left inverse ρ a\rho_a which itself has a left inverse λ a\lambda_a. It follows that ρ a\rho_a is also a right inverse to η a\eta_a, since

ρ aη a =ρ aη aρ aλ a Id =ρ aη aρ a Idλ a =ρ aλ a =Id. \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 aa is WW-local we find that η a:aa¯\eta_a : a \to \bar a is an isomorphism, hence that aa is isomorphic to an object of FF.

Corollary

FF is equivalent to the full subcategory E WlocE_{W-loc} of EE on WW-local objects.

Proof

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

F E E Wloc. \array{ F &&\hookrightarrow&& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow \\ && E_{W-loc} } \,.

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

Proposition

FF is equivalent to the localization E[W 1]E[W^{-1}] of EE at WW.

Proof

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

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

Hom E[W 1](a,b)=colimapWaHom E(a,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

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

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

(ab)(a b Id a a). (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 aa be any object in EE and let η a:aa¯\eta_a : a \to \bar a be the component of the unit of our adjunction on aa, as above. By one of the above propositons, η a\eta_a is in WW. This means that the span

a Id a a η a a¯ \array{ a &\stackrel{Id_a}{\to}& a \\ \downarrow^{\eta_a} \\ \bar a }

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

a η a a¯ Id a a. \array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a } \,.

Since every a¯\bar a is in the image of our functor, this shows that it is essentially surjective.

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

a b w a \array{ a' &\to& b \\ \downarrow^w \\ a }

with wWw \in W has a unique extension

a b w a. \array{ a' &\to& b \\ \downarrow^w & \nearrow \\ a } \,.

But this implies that in the colimit that defines the hom-set of E[W 1]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 Hom E(a,b)Hom F(a,b)Hom_E(a,b) \simeq Hom_F(a,b) so that indeed

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

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

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

Factorizations and images

There is a factorization system on the 2-category Topos whose left class is the surjective geometric morphisms and whose right class is the geometric embeddings. The factorization of a geometric morphism can be said to construct its image in the topos-theoretic sense.

See geometric surjection/embedding factorization.

In the more general context of (∞,1)-topos theory an (,1)(\infty,1)-geometric embedding is an (∞,1)-geometric morphism

(f *f *):𝒳𝒴 (f^* \dashv f_*) : \mathcal{X} \stackrel{\leftarrow}{\hookrightarrow} \mathcal{Y}

such that the right adjoint direct image f *f_* is a full and faithful (∞,1)-functor.

See reflective sub-(∞,1)-category for more details.

References

Section VII, 4 of

and section A4.2 of

Revised on January 14, 2014 11:05:19 by Urs Schreiber (89.204.153.51)