topos theory

Contents

Definition

Local geometric morphism and relative local topos

Definition

A local geometric morphism $f:E\to S$ between toposes $E,S$ is

• $\left({f}^{*}⊣{f}_{*}\right):E\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}S$(f^* \dashv f_*) \colon E \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*} {\to}} S
• such that a further right adjoint ${f}^{!}:S\to E$ exists

$\left({f}^{*}⊣{f}_{*}⊣{f}^{!}\right):E\stackrel{\stackrel{{f}^{*}}{←}}{\stackrel{\underset{{f}_{*}}{\to }}{\underset{{f}^{!}}{←}}}S$(f^* \dashv f_* \dashv f^!) : E \stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\underset{f_*}{\to}}{\underset{f^!}{\leftarrow}}} S
• and such that one, hence all, of the following equivalent conditions hold:

1. The right adjoint ${f}^{!}$ is an $S$-indexed functor.
2. $f$ is connected, i.e. ${f}^{*}$ is fully faithful.
3. The right adjoint ${f}^{!}$ is fully faithful.
4. The right adjoint ${f}^{!}$ is cartesian closed.

When we regard $E$ as a topos over $S$, so that $f$ is regarded as its global section geometric morphism in the category of toposes over $S$, then we say that $E$ is a local $S$-topos. In this case we may label the functors involved as

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

to indicate that if we think of $\Gamma$ as sending a space to its underlying $S$-object of points by forgetting cohesion, then $\mathrm{Disc}$ creates the discrete space/discrete object and $\mathrm{Codisc}$ the codiscrete space/codiscrete object on an object in $S$.

This is especially common when $S=$ Set, in which case the final condition is automatic since all functors are $\mathrm{Set}$-indexed. Hence in that case we have the following simpler definition.

Local topos

Definition

A sheaf topos $𝒯$ is a local topos if the global section geometric morphism $𝒯\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{\Gamma }{\to }}\mathrm{Set}$ has a further right adjoint functor $\left(\mathrm{LConst}⊢\Gamma ⊢\mathrm{CoDisc}\right)$

$\mathrm{CoDisc}:\mathrm{Set}↪𝒯.$CoDisc \colon Set \hookrightarrow \mathcal{T}.

(As just stated, it is automatic in the case over $\mathrm{Set}$ that this is furthermore a full and faithful functor.)

Another way of stating this is that a Grothendieck topos is local if and only if the terminal object $1$ is connected and projective (since this means precisely that $\Gamma =\mathrm{hom}\left(1,-\right)$ preserves colimits, and therefore has a right adjoint by virtue of an adjoint functor theorem). Another term for this: we say $1$ is tiny.

Properties

Equivalent characterizations

Proposition

A geometric morphism $f:E\to S$ is local precisely if

1. there exists a geometric morphism $c:S\to E$ such that $f\circ c\simeq \mathrm{id}$;

2. for every other geometric morphism $g:G\to S$ the composite $c\circ g$ is an initial object in the hom-category ${\mathrm{Topos}}_{/S}\left(g,f\right)$ of the slice 2-category of Topos over $S$.

This is (Johnstone, theorem 3.6.1 vi)).

Remark

In particular this means that the category of topos points of a local topos has a contractible nerve.

General

Proposition

The global section geometric morphism of any local $𝒮$-topos (over a base topos $𝒮$) is a Grothendieck fibration and a Grothendieck opfibration.

This appears in (Shulman).

Proposition

The Freyd cover of a topos is a local topos, and in fact freely so. Every local topos is a retract of a Freyd cover.

This appears as (Johnstone, lemma C3.6.4).

Homotopy dimension

Proposition

In a local sheaf topos over Set, every inhabited object is globally inhabited:

every object $X$ for which the unique morphism $X\to *$ to the terminal object is an epimorphism has a global point $*\to X$.

Proof

Since in a local topos the global section functor $\Gamma$ is a left adjoint, it preserves epimorphisms. Since it is a right adjoint it preserves the terminal object. Therefore $\Gamma \left(X\right)\to \Gamma \left(*\right)\simeq *$ is an epimorphism in Set, hence a surjection, meaning that $\Gamma \left(X\right)$ is inhabited. Since $\Gamma \left(X\right)\simeq \mathrm{Hom}\left(*,X\right)$ (see global section geometric morphism), the claim follows.

Remark

In a topos every epimorphism is an effective epimorphism. Therefore $X\to *$ being an epi means that $X$ is a (-1)-connected object. Therefore the above statement says in terms of (infinity,1)-category theory that a non-trivial local topos has homotopy dimension 0.

The same is true for any local (infinity,1)-topos.

Concrete sheaves

Every local topos $\Gamma :E\to S$ comes with a notion of concrete sheaves, a reflective subcategory ${\mathrm{Conc}}_{\Gamma }\left(E\right)↪E$ which factors the topos inclusion of $S$:

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

and is a quasitopos. See concrete sheaf for details.

Homotopy equivalence

Since a local geometric morphism has a left adjoint in the 2-category Topos, it is necessarily a homotopy equivalence of toposes.

Elementary Axiomatization

For any local topos $\Gamma :ℰ\to 𝒮$, the base topos $𝒮$ is equivalent to the category of sheaves for a Lawvere-Tierney topology $j$ on $ℰ$. A sound and complete elementary axiomatization of local maps of (bounded) toposes can be given in terms of properties of topos $E$ and topology $j$ (AwodeyBirkedal)

We discuss first

and then

themselves.

The setup

Let $ℰ$ be an elementary topos equipped with a Lawvere-Tierney topology $j:\Omega \to \Omega$.

Write $V↦♯V$ for the $j$-closure operation on subobjects $V↪X$, the sharp modality

$\begin{array}{ccc}V& \to & *\\ ↓& & ↓\\ ♯V& \to & & \to & *\\ ↓& & ↓& & ↓\\ X& \stackrel{{\chi }_{V}}{\to }& \Omega & \stackrel{j}{\to }& \Omega \end{array}$\array{ V &\to& {*} \\ \downarrow && \downarrow \\ \sharp V &\to& &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{\chi_V}{\to}& \Omega &\stackrel{j}{\to}& \Omega }

Write

$\left(\mathrm{Disc}⊣\Gamma \right):{\mathrm{Sh}}_{j}\left(ℰ\right)\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{coDisc}}{\to }}ℰ$(Disc \dashv \Gamma) : Sh_j(\mathcal{E}) \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}
Definition

We say that $j$ is an essential topology if for all objects $X$ the closure operation $♯:\mathrm{Sub}\left(X\right)\to \mathrm{Sub}\left(X\right)$ on posets of subobjects has a left adjoint $♭⊣♯$:

$\left(U↪♯V\right)⇔\left(♭U↪V\right)\phantom{\rule{thinmathspace}{0ex}}.$(U \hookrightarrow \sharp V) \Leftrightarrow (\flat U \hookrightarrow V) \,.

This appears under the term “principal” in (Awodey-Birkedal, def. 2.1).

Remark

We use the notation ”$♭$” and ”$♯$” oppositely to the use on p.14 of Awodey-Birkedal. Our convention is such that it harmonizes with the terminology at cohesive topos and cohesive (infinity,1)-topos, where it makes interpretational sense to pronounce ”$♭$” as “flat”.

Observation

The left adjoints $♭:\mathrm{Sub}\left(X\right)\to \mathrm{Sub}\left(X\right)$ for all $X\in ℰ$ extend to a functor $♭:ℰ\to ℰ$ on all of $ℰ$.

(…)

Proposition

A Lawvere-Tierney topology $j$ is essential, $\left(♭⊣♯\right)$, precisely if for all objects $X$ there exists a least $♯$-dense subobject ${U}_{X}↪X$.

This appears as (AwodeyBirkedal, lemma 2.3).

Proof

By the discussion at category of sheaves we have that $♯$ is given by the composite

$♯:ℰ\stackrel{\Gamma }{\to }{\mathrm{Sh}}_{j}\left(ℰ\right)\stackrel{\mathrm{coDisc}}{↪}ℰ\phantom{\rule{thinmathspace}{0ex}},$\sharp : \mathcal{E} \stackrel{\Gamma}{\to} Sh_j(\mathcal{E}) \stackrel{coDisc}{\hookrightarrow} \mathcal{E} \,,

where the first morphism is sheafification and the second is full and faithful.

If now the left adjoint $♭$ exists, it follows that this comes from a left adjoint $\mathrm{Disc}$ to $\Gamma$ as

$♭:ℰ\stackrel{\stackrel{\mathrm{Disc}}{↩}}{\underset{\Gamma }{\to }}{\mathrm{Sh}}_{j}\left(ℰ\right)\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{coDisc}}{↪}}ℰ:♯\phantom{\rule{thinmathspace}{0ex}}.$\flat : \mathcal{E} \stackrel{\overset{Disc}{\hookleftarrow}}{\underset{\Gamma} {\to}} Sh_j(\mathcal{E}) \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}} \mathcal{E} : \sharp \,.

Therefore the $\left(\mathrm{Disc}⊣\Gamma \right)$-counit provides morphisms

$\left(♭X\to X\right)=\left(♭X\to {U}_{x}↪X\right)\phantom{\rule{thinmathspace}{0ex}},$(\flat X \to X) = (\flat X \to U_x \hookrightarrow X) \,,

whose image factorization ${U}_{x}↪X$ we claim provides the least dense subobjects.

To show that ${U}_{X}$ is dense it is sufficient to show that

$♯\left(♭X\to X\right)=\left(\mathrm{coDisc}\Gamma \mathrm{Disc}\Gamma \to \mathrm{coDisc}\Gamma X\right)$\sharp (\flat X \to X) = (coDisc \Gamma Disc \Gamma \to coDisc \Gamma X)

is an isomorphism.

Composing this morphism with $\mathrm{CoDisc}$ of the $\left(\mathrm{Disc}⊣\Gamma \right)$-unit on $\Gamma X$ (which is an isomorphism since $\mathrm{Disc}$ is a full and faithful functor by the discussion at fully faithful adjoint triples) and using the $\left(\mathrm{Disc}⊣\Gamma \right)$ triangle identity we have

$\mathrm{coDisc}\left(\Gamma X\stackrel{\simeq }{\to }\Gamma \mathrm{Disc}\Gamma X\stackrel{}{\to }\Gamma X\right)=\mathrm{coDisc}\left(\Gamma X\stackrel{\mathrm{Id}}{\to }\Gamma X\right)\phantom{\rule{thinmathspace}{0ex}}.$coDisc( \Gamma X \stackrel{\simeq}{\to} \Gamma Disc \Gamma X \stackrel{}{\to} \Gamma X ) = coDisc ( \Gamma X \stackrel{Id}{\to} \Gamma X) \,.

Using that also $\mathrm{coDisc}$ is full and faithful and then 2-out-of-3 for isomorphisms it follows that $\mathrm{coDisc}\Gamma \mathrm{Disc}\Gamma X\stackrel{\simeq }{\to }\mathrm{coDisc}\Gamma X$ hence

$♯♭X\stackrel{\simeq }{\to }♯X$\sharp \flat X \stackrel{\simeq}{\to} \sharp X

is indeed an isomorphism.

Moreover, by one of the equivalent characterizations of reflective subcategories we have (…)

Definition

An object $X\in ℰ$ is called discrete if for all $\Gamma$-local isomorphisms $f:A\to B$ the induced morphism

$ℰ\left(X,A\right)\to ℰ\left(X,B\right)$\mathcal{E}(X,A) \to \mathcal{E}(X,B)

is an isomorphism (of sets, hence a bijection).

Definition

An object $X\in ℰ$ is called o-discrete if $♭X\simeq X$.

Lemma

Every discrete object is o-discrete.

The axioms

Definition

On $ℰ$ a elementary topos with Lawvere-Tierney topology $j$ consider the following axioms.

• Axiom 1. $j$ is essential.

• Axiom 2 a. There is an object $G\in ℰ$ such that every object is a subquotient of the product of a discrete object with $G$.

• Axiom 2 b. With $G$ as above, there is a discrete object $G\prime$ and an epimorphism $G\prime \to ♭G$.

• Axiom 3. For all discrete objects $D$, if the internal hom $\left[X,D\right]$ is o-discrete, then $X$ is also discrete.

• Axiom 4. Discrete objects are closed under binary products.

Theorem

These axioms characterize local geometric morphisms $ℰ\to {\mathrm{Sh}}_{j}\left(ℰ\right)\simeq {D}_{j}\left(ℰ\right)$.

If $G$ is fixed to be the terminal object (in which case Axiom 2 b becomes empty), then they characterize local and localic geometric morphisms.

This is (Awodey-Birkedal, theorem 3.1) together with the discussion around remark 3.7.

Examples

Easy examples

Example

If $C$ is a small category with a terminal object $*\in C$, then the presheaf topos $\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ is a local topos.

Proof

Notice that $\mathrm{Set}\simeq \left[*,\mathrm{Set}\right]$ is the presheaf topos over the point category, the category with a single object and a single morphism. Therefore the constant presheaf functor

$\mathrm{Const}:\mathrm{Set}\to \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$Const : Set \to [C^{op}, Set]

can be thought of as sending a set $S\in \mathrm{Set}$, hence a functor $S:*\to \mathrm{Set}$ to the composite functor

$\mathrm{Const}\left(S\right):C\to *\stackrel{S}{\to }\mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$Const(S) \colon C \to * \stackrel{S}{\to} Set \,.

Notice that in the presence of a terminal object in $C$, $\mathrm{Const}$ is a full and faithful functor: a natural transformation $\mathrm{Const}\left({S}_{1}\right)\to \mathrm{Const}\left({S}_{2}\right)$ has components

$\begin{array}{ccccc}{S}_{1}=& \mathrm{Const}\left({S}_{1}\right)\left(*\right)& \stackrel{f}{\to }& \mathrm{Const}\left({S}_{2}\right)\left(*\right)& ={S}_{2}\\ & {↓}^{\mathrm{id}}& & {↓}^{\mathrm{id}}\\ & \mathrm{Const}\left({S}_{1}\right)\left(U\right)& \to & \mathrm{Const}\left({S}_{2}\right)\left(U\right)\end{array}$\array{ S_1 = & Const(S_1)(*) &\stackrel{f}{\to}& Const(S_2)(*) & = S_2 \\ & \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ & Const(S_1)(U) &\to& Const(S_2)(U) }

where the vertical morphisms are $\mathrm{Const}\left(U\to *\right)$, the point being that they exist for every $U\in C$ given the presence of the terminal object. It follows that such a natural transformation is given by any and one and the same function $f:{S}_{1}\to {S}_{2}$.

The functor $\mathrm{Const}$ has a left adjoint and a right adjoint, and these are – essentially by definition – the colimit and the limit operations

$\left(\underset{\to }{\mathrm{lim}}⊢\mathrm{Const}⊢\underset{←}{\mathrm{lim}}\right)$(\underset{\rightarrow}{\lim} \vdash Const \vdash \underset{\leftarrow}{\lim})

which send a presheaf/functor $F:{C}^{\mathrm{op}}\to \mathrm{Set}$ to its colimit $\underset{\to }{\mathrm{lim}}F\in \mathrm{Set}$ or limit $\underset{←}{\mathrm{lim}}F\in \mathrm{Set}$, respectively.

Since adjoints are essentially unique, it follows that the global section functor $\Gamma :\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ is given by taking the limit, $\Gamma \simeq \underset{←}{\mathrm{lim}}$.

Observe that the terminal object $*\in C$ is the initial object in the opposite category ${C}^{\mathrm{op}}$. But the limit over a diagram with initial object is given simply by evaluation at that object, and so we have for any $F\in \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ that

$\begin{array}{rl}\Gamma \left(F\right)& \simeq \underset{←}{\mathrm{lim}}F\\ & \simeq F\left(*\right)\in \mathrm{Set}\end{array}$\begin{aligned} \Gamma(F) & \simeq \underset{\leftarrow}{\lim} F \\ & \simeq F(*) \in Set \end{aligned}

hence that the global section functor is simply given by evaluating a presheaf on the terminal object of $C$.

Limits and colimit in a presheaf category $\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ are computed objectwise over $C$ (see at limits and colimits by example). Therefore evaluation at any object in $C$ preserves in limits and colimits, and in particular evaluation at the terminal object does. Therefore $\Gamma$ preserves all colimits. Hence by the adjoint functor theorem it has a further right adjoint $\mathrm{CoDisc}$.

We can compute it explicityl by the Yoneda lemma and using the defining Hom-isomorphism of adjoints to be the functor $\mathrm{CoDisc}:\mathrm{Set}\to \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ such that for $S\in \mathrm{Set}$ the presheaf $\mathrm{CoDisc}\left(S\right)$ is given over $U\in C$ by

$\begin{array}{rl}\mathrm{CoDisc}\left(S\right)\left(U\right)& \underset{\mathrm{Yoneda}}{\overset{\simeq }{\to }}{\mathrm{Hom}}_{\left[{C}^{\mathrm{op}},\mathrm{Set}\right]}\left(U,\mathrm{CoDisc}\left(S\right)\right)\\ & \underset{\mathrm{adjunction}}{\overset{\simeq }{\to }}{\mathrm{Hom}}_{\mathrm{Set}}\left(\Gamma \left(U\right),S\right)\phantom{\rule{thinmathspace}{0ex}}.\end{array}$\begin{aligned} CoDisc(S)(U) & \underoverset{Yoneda}{\simeq}{\to} Hom_{[C^{op}, Set]}(U,CoDisc(S)) \\ & \underoverset{adjunction}{\simeq}{\to} Hom_{Set}(\Gamma(U), S) \,. \end{aligned}

So in conclusion we have an adjoint triple $\left(\mathrm{Const}⊢\Gamma ⊢\mathrm{CoDisc}\right)$ where $\mathrm{Const}$ is a full and faithful functor. By the discussion at fully faithful adjoint triples it follows then that also $\mathrm{CoDisc}$ is full and faithful.

Proposition

The converse to prop. 1 is true if $C$ is Cauchy complete.

Example

If $X$ is a topological space, or more generally a locale, then $\mathrm{Sh}\left(X\right)$ is local (over Set) iff $X$ has a focal point, i.e. a point whose only neighborhood is the whole space.

Sheaves on a local site

For $C$ a local site, the category of sheaves $\mathrm{Sh}\left(C\right)$ is a local topos over $\mathrm{Set}$.

For instance CartSp is a local site. Objects in $\mathrm{Sh}\left(C\right)$ are generalized smooth spaces such as diffeological spaces. The further right adjoint

$\mathrm{Codisc}:\mathrm{Set}\to \mathrm{Sh}\left(\mathrm{CartSp}\right)$Codisc : Set \to Sh(CartSp)

is the functor that sends a set to the diffeological space on that set with codiscrete smooth structure (every map of sets is smooth).

Relative Realizability

Let $A$ be a partial combinatory algebra and let $A♯\subseteq A$ be a sub partial combinatory algebra of $A$. Then there is a (localic) local geometric morphism from the relative realizability topos? $\mathrm{RT}\left(A,A♯\right)$ to the standard realizability topos $\mathrm{RT}\left(A♯\right)$.

Localization

Let $\mathrm{LocTopos}$ denote the 2-category of local Grothendieck toposes (over Set) with all geometric morphisms between them. Let $\mathrm{PTopos}$ denote the 2-category whose objects are pointed toposes? (i.e. (Grothendieck) toposes $E$ equipped with a geometric morphism $s:\mathrm{Set}\to E$), and whose morphisms are pairs $\left(f,\alpha \right)$ such that $f:E\to E\prime$ is a geometric morphism and $\alpha :s\prime \to fs$ is a (not necessarily invertible) geometric transformation.

Note that if $E$ is a local topos with global sections geometric morphism ${e}^{*}⊣{e}_{*}$, then the adjunction ${e}_{*}⊣{e}^{!}$ is also a geometric morphism $\mathrm{Set}\to E$. In this way we have a functor $\mathrm{LocTopos}\to \mathrm{PTopos}$, which is a full embedding, and turns out to have a right adjoint: this right adjoint is called the localization of a pointed topos at its specified point. For example:

Proposition

If $C$ is a small category and $U$ is an object of $C$, then the localization of the presheaf topos $\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ at the point induced by $U:1\to C$ can be identified with the presheaf topos $\left[\left(C/U{\right)}^{\mathrm{op}},\mathrm{Set}\right]$ over the over category of $C$ over $U$. By the general properties of over toposes, this is equivalently the over-topos $\mathrm{PSh}\left(C\right)/U$ (where $U$ is regarded in $\mathrm{PSh}\left(C\right)$ by the Yoneda embedding).

Proposition

If $X=\mathrm{Spec}\left(A\right)$ is the Zariski spectrum of a commutative ring $A$, and $P\subset A$ is a prime ideal of $A$ (i.e. a point of $X$), then the localization of $\mathrm{Sh}\left(X\right)$ at $P:1\to X$ can be identified with $\mathrm{Sh}\left(\mathrm{Spec}\left({A}_{P}\right)\right)$, where ${A}_{P}$ denotes the localization of $A$ at $P$. Of course, this is the origin of the terminology.

A similar construction is possible for bounded toposes over any base (not just Set).

Local over-toposes

Proposition

For $ℰ$ a Grothendieck topos and $X\in ℰ$ an object, the over topos $ℰ/X$ is local if $X$ is a tiny object.

Proof

We check that the global section geometric morphism $\Gamma :ℰ/X\to \mathrm{Set}$ preserves colimits. It is given by the hom-functor out of the terminal object of $ℰ/X$, which is $\left(X\stackrel{\mathrm{Id}}{\to }X\right)$:

$\Gamma :\left(A\stackrel{f}{\to }X\right)↦{\mathrm{Hom}}_{ℰ/X}\left({\mathrm{Id}}_{X},f\right)\phantom{\rule{thinmathspace}{0ex}}.$\Gamma : (A \stackrel{f}{\to} X) \mapsto Hom_{\mathcal{E}/X}(Id_X, f) \,.

The hom-sets in the over category are fibers of the hom-sets in $ℰ$: we have a pullback diagram

$\begin{array}{ccc}{\mathrm{Hom}}_{ℰ/X}\left({\mathrm{Id}}_{X},\left(A\to X\right)\right)& \to & {\mathrm{Hom}}_{ℰ}\left(X,A\right)\\ ↓& & {↓}^{{f}_{*}}\\ *& \stackrel{{\mathrm{Id}}_{x}}{\to }& {\mathrm{Hom}}_{ℰ}\left(X,X\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ Hom_{\mathcal{E}/X}(Id_X, (A \to X)) &\to& Hom_{\mathcal{E}}(X,A) \\ \downarrow && \downarrow^{f_*} \\ * &\stackrel{Id_x}{\to}& Hom_{\mathcal{E}}(X,X) } \,.

Moreover, overserve that colimits in the over category are computed in $ℰ$.

${\underset{\to }{\mathrm{lim}}}_{i}\left({A}_{i}\stackrel{{f}_{i}}{\to }X\right)\simeq \left({\underset{\to }{\mathrm{lim}}}_{i}{A}_{i}\right)\to X\phantom{\rule{thinmathspace}{0ex}}.${\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X) \simeq ({\lim_\to}_i A_i) \to X \,.

If $X$ is a tiny object then by definition we have

${\mathrm{Hom}}_{ℰ}\left(X,{\mathrm{lim}}_{i}{A}_{i}\right)\simeq {\underset{\to }{\mathrm{lim}}}_{i}{\mathrm{Hom}}_{ℰ}\left(X,{A}_{i}\right)\phantom{\rule{thinmathspace}{0ex}},$Hom_{\mathcal{E}}(X, {\lim}_i A_i) \simeq {\lim_\to}_i Hom_{\mathcal{E}}(X, A_i) \,,

Inserting all this into the above pullback gives the pullback

$\begin{array}{ccc}{\mathrm{Hom}}_{ℰ/X}\left({\mathrm{Id}}_{X},{\underset{\to }{\mathrm{lim}}}_{i}\left({A}_{i}\to X\right)\right)& \to & {\underset{\to }{\mathrm{lim}}}_{i}{\mathrm{Hom}}_{ℰ}\left(X,{A}_{i}\right)\\ ↓& & {↓}^{{f}_{*}}\\ *& \stackrel{{\mathrm{Id}}_{x}}{\to }& {\mathrm{Hom}}_{ℰ}\left(X,X\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ Hom_{\mathcal{E}/X}(Id_X, {\lim_\to}_i (A_i \to X)) &\to& {\lim_\to}_i Hom_{\mathcal{E}}(X, A_i) \\ \downarrow && \downarrow^{f_*} \\ * &\stackrel{Id_x}{\to}& Hom_{\mathcal{E}}(X,X) } \,.

By universal colimits in the topos Set, this pullback of a colimit is the colimit of the separate pullbacks, so that

$\Gamma \left({\underset{\to }{\mathrm{lim}}}_{i}\left({A}_{i}\to X\right)\right)\right)\simeq {\mathrm{Hom}}_{ℰ/X}\left({\mathrm{Id}}_{X},{\underset{\to }{\mathrm{lim}}}_{i}\left({A}_{i}\to X\right)\right)\simeq {\underset{\to }{\mathrm{lim}}}_{i}{\mathrm{Hom}}_{ℰ/X}\left({\mathrm{Id}}_{X},\left({A}_{i}\to X\right)\right)\simeq {\underset{\to }{\mathrm{lim}}}_{i}\Gamma \left({A}_{i}\to X\right)\phantom{\rule{thinmathspace}{0ex}}.$\Gamma({\lim_\to}_i (A_i \to X))) \simeq Hom_{\mathcal{E}/X}(Id_X, {\lim_\to}_i (A_i \to X)) \simeq {\lim_\to}_i Hom_{\mathcal{E}/X}(Id_X,(A_i \to X)) \simeq {\lim_\to}_i \Gamma(A_i \to X) \,.

So $\Gamma$ does commute with colimits if $X$ is tiny. By the adjoint functor theorem then the right adjoint $\nabla :\mathrm{Set}\to ℰ/X$ does exist and so $ℰ/X$ is a local topos.

Remark

As a special case this reproduces the above statement that slices $\mathrm{PSh}\left(C\right)/j\left(U\right)$ of presheaf toposes over objects in the image of the Yoneda embedding are local: every representable functor is tiny (see there).

Gros toposes

Let $A$ be a commutative ring (such as $ℤ$ or a field), let $\mathrm{Spec}A$ be the prime spectrum of $A$, and let ${𝒵}_{A}$ be the big Zariski topos for $A$ (i.e. the classifying topos for local $A$-algebras). For each element $a$ of $A$, we have an open subset $D\left(a\right)=\left\{𝔭\in \mathrm{Spec}A:a\notin 𝔭\right\}$, and these open subsets constitute a basis for the topology on $\mathrm{Spec}A$. The full subcategory of the frame of open subsets of $\mathrm{Spec}A$ spanned by these basic open subsets admits a contravariant full embedding in the category of finitely-presented $A$-algebras via the functor $D\left(a\right)↦A\left[{a}^{-1}\right]$ (the well-definedness of this functor requires a non-trivial check!), and this functor moreover has the cover lifting property, so induces a local geometric morphism ${𝒵}_{A}\to \mathrm{Sh}\left(\mathrm{Spec}A\right)$.

and

References

Standard references include

and Chapter C3.6 of

A completely internal characterization of local toposes is discussed in

This is based on part 2 of

• Lars Birkedal, Developing Theories of Types and Computability via Realizability PhD Thesis (pdf)

Free local constructions are considered in

Revised on June 5, 2013 01:46:35 by Zhen Lin (131.111.228.88)