topos theory

# Contents

## Idea

When regarding a sheaf as a space defined by how it is probed by test spaces, a concrete sheaf is a generalized space that has (at least) an underlying set of points out of which it is built.

So a concrete sheaf models a space that is given by a set of points and a choice of which morphisms of sets from concrete test spaces into it count as “structure preserving” (e.g. count as smooth, when the sheaf models a smooth space).

More in the intrinsic language of sheaves, a concrete sheaf is a sheaf on a concrete site $C$ that, while perhaps not representable, is “quasi-representable” in that it is a subobject of a sheaf of the form

$U↦{\mathrm{Hom}}_{\mathrm{Set}}\left(\mid U\mid ,S\right)$U \mapsto Hom_{Set}(|U|, S)

where $S$ is a set and $\mid U\mid$ is the set $\mid U\mid :={\mathrm{Hom}}_{C}\left(*,U\right)$ of points underlying the object $U$ in the concrete site $C$.

## Definition

We discuss two definitions: the first one is more elementary and describes concrete sheaves explicitly in terms of properties of the underlying site.

The second one is more abstract and more general, and describes them entirely topos theoretically.

### On a concrete site

###### Definition

A concrete site is a site $C$ with a terminal object $*$ such that

1. the functor ${\mathrm{Hom}}_{C}\left(*,-\right):C\to \mathrm{Set}$ is a faithful functor;

2. for every covering family $\left\{{f}_{i}:{U}_{i}\to U\right\}$ in $C$ the morphism

$\coprod _{i}{\mathrm{Hom}}_{C}\left(*,f\right):\coprod _{i}{\mathrm{Hom}}_{C}\left(*,{U}_{i}\right)\to {\mathrm{Hom}}_{C}\left(*,U\right)$\coprod_i Hom_C(*,f) : \coprod_i Hom_C(*, U_i) \to Hom_C(*, U)

is surjective.

For $X\in \mathrm{PSh}\left(C\right)$ any presheaf, write

${\stackrel{˜}{X}}_{U}:X\left(U\right)\to {\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{Hom}}_{C}\left(*,U\right),X\left(*\right)\right)$\tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))

for the adjunct of the restriction map

$X\left(U\right)×{\mathrm{Hom}}_{C}\left(*,U\right)\to X\left(*\right)\phantom{\rule{thinmathspace}{0ex}},$X(U) \times Hom_C(*,U) \to X(*) \,,

which in turn is the adjunct of the component map of the functor

${X}_{*,U}.{\mathrm{Hom}}_{C}\left(*,U\right)\to {\mathrm{Hom}}_{\mathrm{Set}}\left(X\left(U\right),X\left(*\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$X_{*,U} . Hom_C(*,U) \to Hom_{Set}(X(U), X(*)) \,.
###### Definition

A presheaf $X:{C}^{\mathrm{op}}\to \mathrm{Set}$ on a concrete site is a concrete presheaf if for each $U\in C$ the map ${\stackrel{˜}{X}}_{U}:X\left(U\right)\to {\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{Hom}}_{C}\left(*,U\right),X\left(*\right)\right)$ is injective.

A concrete sheaf is a presheaf that is both concrete and a sheaf.

So a concrete presheaf $X$ is a subobject of the presheaf $U↦{\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{Hom}}_{C}\left(*,U\right),X\left(*\right)\right)$.

Write $\mathrm{Conc}\left(\mathrm{Sh}\left(C\right)\right)↪\mathrm{Sh}\left(C\right)$ for the full subcategory of the category of sheaves on concrete sheaves.

### In a local topos

A more abstract perspective on the previous definition is obtained by noticing the following.

###### Lemma

The category of sheaves on a concrete site is a local topos.

###### Proof

Taking $U=*$ in the second condition defining a concrete site implies that any covering family of $*$ contains a split epimorphism, or equivalently that the only covering sieve of $*$ is the maximal sieve consisting of all morphisms with target $*$. This means that a concrete site is in particular a local site, which implies that its topos of sheaves is a local topos.

In fact, we can formulate the definition of concrete sheaf inside any local topos $E$ over any base topos $S$:

###### Definition

Let

$\left(\mathrm{Disc}⊣\Gamma ⊣\mathrm{Codisc}\right):E\stackrel{\stackrel{\stackrel{\mathrm{Disc}}{←}}{\underset{\Gamma }{\to }}}{\underset{\mathrm{Codisc}}{←}}S$(Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} S

be a local geometric morphism. Since then by definition $S\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{Codisc}}{↪}}E$ is a subtopos the morphisms $V={\Gamma }^{-1}\left(\mathrm{isos}\left(S\right)\right)\subset \mathrm{Mor}E$ that are inverted by $\Gamma$ are the local isomorphisms with respect to which the objects of $S$ are sheaves/$V$-local objects in $E$.

The concrete sheaves are the objects of $E$ that are the $V$-separated objects.

###### Proposition

For $E=\mathrm{Sh}\left(C\right)\stackrel{\Gamma =\mathrm{Hom}\left(*,-\right)}{\to }\mathrm{Set}$ the category of sheaves on a concrete site, this is equivalent to the previous definition.

###### Proof

Since $C$ is concrete, in the global sections geometric morphism $\left(\mathrm{Disc},\Gamma \right):\mathrm{Sh}\left(C\right)\to \mathrm{Set}$, the direct image $\Gamma$ is evaluation on the point: $X↦X\left(*\right)$. The further right adjoint $\mathrm{Codisc}:\mathrm{Set}\to \mathrm{Sh}\left(C\right)$, sends a set $A$ to the functor $U↦{\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{Hom}}_{C}\left(*,U\right),A\right)$. Moreover, this right adjoint $\mathrm{Codisc}$ is fully faithful and thus embeds $\mathrm{Set}$ as a subtopos of $\mathrm{Sh}\left(C\right)$.

We observe that $\left(\Gamma ⊣\mathrm{Codisc}\right):\mathrm{Set}\to \mathrm{Sh}\left(C\right)$ is the localization of $\mathrm{Sh}\left(C\right)$ at the set $\left\{\mathrm{Disc}\Gamma U\to U\mid U\in C\right\}$ of counits of the adjunction $\left(\mathrm{Disc}⊣\Gamma \right)$ on representables: because if for $X\in \mathrm{Sh}\left(C\right)$ we have that

$\begin{array}{rl}{\mathrm{Hom}}_{\mathrm{Sh}\left(C\right)}\left(\mathrm{Disc}\Gamma U\to U,X\right)& =\left(X\left(U\right)\to {\mathrm{Hom}}_{\mathrm{Set}}\left(\Gamma \left(U\right),\Gamma \left(X\right)\right)\right)\\ & =\left(X\left(U\right)\to {\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{Hom}}_{C}\left(*,U\right)\right),X\left(*\right)\right)\end{array}$\begin{aligned} Hom_{Sh(C)}(Disc \Gamma U \to U, X) & = (X(U) \to Hom_{Set}(\Gamma(U), \Gamma(X))) \\ & = (X(U) \to Hom_{Set}(Hom_C(*,U)), X(*)) \end{aligned}

is an isomorphism, then clearly $X=\mathrm{Codisc}\left(X\left(*\right)\right)$.

On the other hand, comparison with the previous definition shows that this is a monomorphism precisely if $X$ is a concrete sheaf. But this is also the definition of a separated object.

So the concrete sheaves on $C$ are precisely the separated objects for this Lawvere-Tierney topology on $\mathrm{Sh}\left(C\right)$ that corresponds to the subtopos $\mathrm{Codisc}:S↪\mathrm{Sh}\left(C\right)$.

## Properties

### General

Let $\Gamma :E\to S$ be a local topos. From the definition of concrete sheaves as separated presheaves it follows immediately that

###### Proposition

The category of concrete sheaves ${\mathrm{Conc}}_{\Gamma }\left(E\right)$ forms a reflective subcategory of $E$

$S\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{Codisc}}{↪}}{\mathrm{Conc}}_{\Gamma }\left(E\right)\stackrel{\stackrel{\mathrm{Conc}}{←}}{↪}E$S \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} Conc_\Gamma(E) \stackrel{\overset{Conc}{\leftarrow}}{\hookrightarrow} E

which is a quasitopos.

The left adjoint $\mathrm{Conc}$ is concretization which sends a sheaf $X$ to the image sheaf

$\mathrm{Conc}X:U↦\mathrm{Im}\left(X\left(U\right)\to {\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{Hom}}_{C}\left(*,U\right),X\left(*\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$Conc X : U \mapsto Im(X(U) \to Hom_{Set}(Hom_C(*,U), X(*))) \,.

### Slice topos over a concrete object

Let $\left(\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}\right):ℰ\to 𝒮$ be a Grothendieck topos that is a local topos over $𝒮$ and let $X\in ℰ$ be a concrete object, equivalently an object such that the $\left(\Gamma ⊣\mathrm{coDisc}\right)$-counit $X\to \mathrm{coDisc}\Gamma U$ is a monomorphism.

We discuss properties of the over-topos $ℰ/X$.

Notice that

${e}_{0}:=\left(\Gamma ⊣\mathrm{coDisc}\right):𝒮\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{coDisc}}{\to }}ℰ$e_0 := (\Gamma \dashv coDisc) : \mathcal{S} \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}

is the canonical topos point of $ℰ$.

###### Observation

For every global element $\left(x\in \Gamma \left(X\right)\right):*\to X$ (for every $X\in ℰ$) there is a topos point of the form

$\left({e}_{0},x\right):𝒮\stackrel{\stackrel{{x}^{*}}{←}}{\underset{{x}_{*}}{\to }}𝒮/\Gamma \left(X\right)\stackrel{\stackrel{\Gamma /X}{←}}{\underset{\mathrm{coDisc}/X}{\to }}ℰ/X\phantom{\rule{thinmathspace}{0ex}}.$(e_0,x) : \mathcal{S} \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} \mathcal{S}/\Gamma(X) \stackrel{\overset{\Gamma/X}{\leftarrow}}{\underset{coDisc/X}{\to}} \mathcal{E}/X \,.

This is discussed in detail at over-topos -- points.

###### Observation

(relative concretization)

Let $X\in ℰ$ be concrete. Then the image under the $\mathrm{coDisc}/X\circ \Gamma /X$-monad of any object $\left(A\to X\right)\in ℰ/X$ is an object $\left(\stackrel{˜}{A}\to X\right)$ with $\stackrel{˜}{A}$ being concrete.

This $\stackrel{˜}{A}$ is the finest concrete sheaf structure on $\Gamma A$ that extends $\Gamma A\to \Gamma X$ to a morphism of concrete sheaves.

###### Proof

By definition of the slice geometric morphism we have that $\mathrm{coDisc}/X\circ \Gamma /X\left(A\stackrel{f}{\to }X\right)$ is the pullback $\stackrel{˜}{A}\to X$ in

$\begin{array}{ccc}\stackrel{˜}{A}& \to & \mathrm{coDisc}\Gamma A\\ ↓& & {↓}^{\mathrm{coDisc}\Gamma A}\\ X& \to & \mathrm{coDisc}\Gamma X\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \tilde A &\to & coDisc \Gamma A \\ \downarrow && \downarrow^{\mathrlap{coDisc \Gamma A}} \\ X &\to& coDisc \Gamma X } \,,

where the bottom morphism is the $\left(\Gamma ⊣\mathrm{coDisc}\right)$-unit. Since this is a monomorphism by assumption on $X$ it follows that $\stackrel{˜}{A}\to \mathrm{coDisc}\Gamma A$ is a monomorphism. Since $\mathrm{coDisc}$ is a full and faithful functor by assumption on $ℰ$ and $\Gamma$ is a right adjoint it follows that the adjunct $\Gamma \stackrel{˜}{A}\to \Gamma \mathrm{coDisc}\Gamma A\stackrel{\simeq }{\to }\Gamma A$ is a monomorphism, as is its image $\mathrm{coDisc}\Gamma \stackrel{˜}{A}\to \mathrm{coDisc}\Gamma A$ under the right adjoint $\mathrm{coDisc}$.

Then by the universal property of the unit we have a commuting diagram

$\begin{array}{ccc}& & \mathrm{coDisc}\Gamma \stackrel{˜}{A}\\ & ↗& ↓\\ \stackrel{˜}{A}& \to & \mathrm{coDisc}\Gamma A\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ && coDisc \Gamma \tilde A \\ & \nearrow & \downarrow \\ \tilde A &\to& coDisc \Gamma A } \,,

where the bottom and the right morphisms are monomorphisms. Therefore also the diagonal morphism, the $\left(\Gamma ⊣\mathrm{coDisc}\right)$-unit on $\stackrel{˜}{A}$, is a monomorphism, and hence $\stackrel{˜}{A}$ is concrete.

## References

The notion of quasitoposes of concrete sheaves goes back to

• Eduardo Dubuc, Concrete quasitopoi , Lecture Notes in Math. 753 (1979), 239–254

and is further developed in

A review of categories of concrete sheaves, with special attention to sheaves on CartSp, i.e. to diffeological spaces is in

The characterization of concrete sheaves in terms of the extra right adjoint of a local topos originated in discussion with David Carchedi.

Revised on November 23, 2011 17:49:53 by Urs Schreiber (131.174.40.49)