nLab
local geometric morphism

Definition

Local geometric morphism and relative local topos

Definition

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

• a geometric morphism

  $$(f^{*}⊣f_{*}):E\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}S$$(f^* \dashv f_*) \colon E \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*} {\to}} S

• such that a further right adjoint $f^{!}:S\to E$ exists

  $$(f^{*}⊣f_{*}⊣f^{!}):E\stackrel{\stackrel{f^{*}}{\leftarrow }}{\stackrel{\underset{f_{*}}{\to }}{\underset{f^{!}}{\leftarrow }}}S$$(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

$$(\mathrm{Disc}⊣\Gamma ⊣\mathrm{Codisc}):E\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\stackrel{\underset{\Gamma }{\to }}{\underset{\mathrm{Codisc}}{\leftarrow }}}S$$(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 $𝒯\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{Set}$ has a further right adjoint functor $(\mathrm{LConst}⊢\Gamma ⊢\mathrm{CoDisc})$

$$\mathrm{CoDisc}:\mathrm{Set}\hookrightarrow 𝒯.$$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 $\Gamma =\mathrm{hom}(1,-)$ 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 ${\mathrm{Topos}}_{/S}(g,f)$ 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 $\Gamma (X)\to \Gamma (*)\simeq *$ is an epimorphism in Set, hence a surjection, meaning that $\Gamma (X)$ is inhabited. Since $\Gamma (X)\simeq \mathrm{Hom}(*,X)$ (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 ${\mathrm{Conc}}_{\Gamma }(E)\hookrightarrow E$ which factors the topos inclusion of $S$:

$$S\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{Codisc}}{\hookrightarrow }}{\mathrm{Conc}}_{\Gamma }(E)\stackrel{\stackrel{\mathrm{Conc}}{\leftarrow }}{\hookrightarrow }E$$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

• the setup

and then

• the axioms

themselves.

The setup

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

Write $V\mapsto ♯V$ for the $j$-closure operation on subobjects $V\hookrightarrow X$, the sharp modality

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

Write

$$(\mathrm{Disc}⊣\Gamma ):{\mathrm{Sh}}_j(ℰ)\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{coDisc}}{\to }}ℰ$$(Disc \dashv \Gamma) : Sh_j(\mathcal{E}) \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}

for the reflective subcategory of j-sheaves.

Definition

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

$$(U\hookrightarrow ♯V)\iff (♭U\hookrightarrow V).$$(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 $♭:\mathrm{Sub}(X)\to \mathrm{Sub}(X)$ for all $X\in ℰ$ extend to a functor $♭:ℰ\to ℰ$ on all of $ℰ$.

Proof

(…)

Proposition

A Lawvere-Tierney topology $j$ is essential, $(♭⊣♯)$, precisely if for all objects $X$ there exists a least $♯$-dense subobject $U_X\hookrightarrow X$.

This appears as (AwodeyBirkedal, lemma 2.3).

Proof

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

$$♯:ℰ\stackrel{\Gamma }{\to }{\mathrm{Sh}}_j(ℰ)\stackrel{\mathrm{coDisc}}{\hookrightarrow }ℰ,$$\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

$$♭:ℰ\stackrel{\stackrel{\mathrm{Disc}}{\hookleftarrow }}{\underset{\Gamma }{\to }}{\mathrm{Sh}}_j(ℰ)\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{coDisc}}{\hookrightarrow }}ℰ:♯.$$\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 $(\mathrm{Disc}⊣\Gamma )$-counit provides morphisms

$$(♭X\to X)=(♭X\to U_x\hookrightarrow X),$$(\flat X \to X) = (\flat X \to U_x \hookrightarrow X) \,,

whose image factorization $U_x\hookrightarrow X$ we claim provides the least dense subobjects.

To show that $U_X$ is dense it is sufficient to show that

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

is an isomorphism.

Composing this morphism with $\mathrm{CoDisc}$ of the $(\mathrm{Disc}⊣\Gamma )$-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 $(\mathrm{Disc}⊣\Gamma )$ triangle identity we have

$$\mathrm{coDisc}(\Gamma X\stackrel{\simeq }{\to }\Gamma \mathrm{Disc}\Gamma X\stackrel{}{\to }\Gamma X)=\mathrm{coDisc}(\Gamma X\stackrel{\mathrm{Id}}{\to }\Gamma X).$$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

$$ℰ(X,A)\to ℰ(X,B)$$\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 $[X,D]$ is o-discrete, then $X$ is also discrete.

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

Theorem

These axioms characterize local geometric morphisms $ℰ\to {\mathrm{Sh}}_j(ℰ)\simeq D_j(ℰ)$.

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 $[C^{\mathrm{op}},\mathrm{Set}]$ is a local topos.

Proof

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

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

$$\mathrm{Const}(S):C\to *\stackrel{S}{\to }\mathrm{Set}.$$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 $\mathrm{Const}(S_1)\to \mathrm{Const}(S_2)$ has components

$$\begin{array}{ccccc}{S}_1=& \mathrm{Const}(S_1)(*)& \stackrel{f}{\to }& \mathrm{Const}(S_2)(*)& =S_2\\ & {\downarrow }^{\mathrm{id}}& & {\downarrow }^{\mathrm{id}}\\ & \mathrm{Const}(S_1)(U)& \to & \mathrm{Const}(S_2)(U)\end{array}$$\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 $\mathrm{Const}(U\to *)$, 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

$$(\underrightarrow{\lim }⊢\mathrm{Const}⊢\underleftarrow{\lim })$$(\underset{\rightarrow}{\lim} \vdash Const \vdash \underset{\leftarrow}{\lim})

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

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

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 $F\in [C^{\mathrm{op}},\mathrm{Set}]$ that

$$\begin{array}{rl}\Gamma (F)& \simeq \underleftarrow{\lim }F\\ & \simeq F(*)\in \mathrm{Set}\end{array}$$\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 $[C^{\mathrm{op}},\mathrm{Set}]$ 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 $\mathrm{CoDisc}:\mathrm{Set}\to [C^{\mathrm{op}},\mathrm{Set}]$ such that for $S\in \mathrm{Set}$ the presheaf $\mathrm{CoDisc}(S)$ is given over $U\in C$ by

$$\begin{array}{rl}\mathrm{CoDisc}(S)(U)& \underset{\mathrm{Yoneda}}{\overset{\simeq }{\to }}{\mathrm{Hom}}_{[C^{\mathrm{op}},\mathrm{Set}]}(U,\mathrm{CoDisc}(S))\\ & \underset{\mathrm{adjunction}}{\overset{\simeq }{\to }}{\mathrm{Hom}}_{\mathrm{Set}}(\Gamma (U),S).\end{array}$$\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 $(\mathrm{Const}⊢\Gamma ⊢\mathrm{CoDisc})$ 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 $\mathrm{Sh}(X)$ 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 $\mathrm{Sh}(C)$ is a local topos over $\mathrm{Set}$.

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

$$\mathrm{Codisc}:\mathrm{Set}\to \mathrm{Sh}(\mathrm{CartSp})$$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? $\mathrm{RT}(A,A♯)$ to the standard realizability topos $\mathrm{RT}(A♯)$.

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

Proposition

If $X=\mathrm{Spec}(A)$ 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 $\mathrm{Sh}(X)$ at $P:1\to X$ can be identified with $\mathrm{Sh}(\mathrm{Spec}(A_P))$, 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 $(X\stackrel{\mathrm{Id}}{\to }X)$:

$$\Gamma :(A\stackrel{f}{\to }X)\mapsto {\mathrm{Hom}}_{ℰ/X}({\mathrm{Id}}_X,f).$$\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

$$\begin{array}{ccc}{\mathrm{Hom}}_{ℰ/X}({\mathrm{Id}}_X,(A\to X))& \to & {\mathrm{Hom}}_{ℰ}(X,A)\\ \downarrow & & {\downarrow }^{f_{*}}\\ *& \stackrel{{\mathrm{Id}}_x}{\to }& {\mathrm{Hom}}_{ℰ}(X,X)\end{array}.$$\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 $ℰ$.

$${\underrightarrow{\lim }}_i(A_i\stackrel{f_i}{\to }X)\simeq ({\underrightarrow{\lim }}_i{A}_i)\to X.$${\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

$${\mathrm{Hom}}_{ℰ}(X,{\lim }_i{A}_i)\simeq {\underrightarrow{\lim }}_i{\mathrm{Hom}}_{ℰ}(X,A_i),$$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

$$\begin{array}{ccc}{\mathrm{Hom}}_{ℰ/X}({\mathrm{Id}}_X,{\underrightarrow{\lim }}_i(A_i\to X))& \to & {\underrightarrow{\lim }}_i{\mathrm{Hom}}_{ℰ}(X,A_i)\\ \downarrow & & {\downarrow }^{f_{*}}\\ *& \stackrel{{\mathrm{Id}}_x}{\to }& {\mathrm{Hom}}_{ℰ}(X,X)\end{array}.$$\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

$$\Gamma ({\underrightarrow{\lim }}_i(A_i\to X)))\simeq {\mathrm{Hom}}_{ℰ/X}({\mathrm{Id}}_X,{\underrightarrow{\lim }}_i(A_i\to X))\simeq {\underrightarrow{\lim }}_i{\mathrm{Hom}}_{ℰ/X}({\mathrm{Id}}_X,(A_i\to X))\simeq {\underrightarrow{\lim }}_i\Gamma (A_i\to X).$$\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 $\mathrm{PSh}(C)/j(U)$ 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 $\mathbb{Z}$ 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 $D(a)=\{𝔭\in \mathrm{Spec}A:a\notin 𝔭\}$, 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 $D(a)\mapsto A[a^{-1}]$ (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 ${𝒵}_A\to \mathrm{Sh}(\mathrm{Spec}A)$.

Related concepts

• essential geometric morphism

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos

  • Pi modality $⊣$ flat modality $⊣$ sharp modality

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site, locally ∞-connected site

• connected site

• local site

• cohesive site, (∞,1)-cohesive site

References

Standard references include

• Peter Johnstone, Ieke Moerdijk, Local maps of toposes Proc. London Math. Soc. (1989) s3-58 (2): 281-305. (pdf)

and Chapter C3.6 of

• Peter Johnstone, Sketches of an Elephant

A completely internal characterization of local toposes is discussed in

• Steve Awodey, Lars Birkedal, Elementary axioms for local maps of toposes, Journal of Pure and Applied Algebra, 177(3):215-230, (2003) (ps, pdf )

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

• Mike Shulman, Discreteness, Concreteness, Fibrations, and Scones (blog post)