# nLab local geometric morphism

### Context

#### Topos Theory

Could not include topos theory - contents

# Contents

## Definition

### Local geometric morphism and relative local topos

###### Definition

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

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

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

$(Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\underset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} S$

to indicate that if we think of $\Gamma$ as sending a space to its underlying $S$-object of points by forgetting cohesion, then $Disc$ creates the discrete space/discrete object and $Codisc$ 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 $Set$-indexed. Hence in that case we have the following simpler definition.

### Local topos

###### Definition

A sheaf topos $\mathcal{T}$ is a local topos if the global section geometric morphism $\mathcal{T} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Set$ has a further right adjoint $CoDisc$, making an adjoint triple $(LConst \vdash \Gamma \vdash CoDisc)$

$CoDisc \colon Set \hookrightarrow \mathcal{T}.$

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

###### Remark

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 $\Gamma = \hom(1, -)$ preserves colimits, and therefore has a right adjoint by virtue of an adjoint functor theorem). Another term for this: we say $1$ is tiny (atomic). Notice the similarity to the concept of amazing right adjoint (the difference being that this is a right adjoint not to the external but to the internal hom out of 1.)

## 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 $f \circ c \simeq id$;

2. for every other geometric morphism $g : G \to S$ the composite $c\circ g$ is an initial object in the hom-category $Topos_{/S}(g,f)$ 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 $\mathcal{S}$-topos (over a base topos $\mathcal{S}$) 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 $\Gamma(X) \simeq Hom(*,X)$ (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 $Conc_\Gamma(E) \hookrightarrow E$ which factors the topos inclusion of $S$:

$S \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} Conc_\Gamma(E) \stackrel{\overset{Conc}{\leftarrow}}{\hookrightarrow} E$

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 $\Gamma \colon \mathcal{E} \to \mathcal{S}$, the base topos $\mathcal{S}$ is equivalent to the category of sheaves for a Lawvere-Tierney topology $j$ on $\mathcal{E}$. 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 $\mathcal{E}$ be an elementary topos equipped with a Lawvere-Tierney topology $j : \Omega \to \Omega$.

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

$\array{ V &\to& {*} \\ \downarrow && \downarrow \\ \sharp V &\to& &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{\chi_V}{\to}& \Omega &\stackrel{j}{\to}& \Omega }$

Write

$(Disc \dashv \Gamma) : Sh_j(\mathcal{E}) \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}$
###### Definition

We say that $j$ is an essential topology if for all objects $X$ the closure operation $\sharp : Sub(X) \to Sub(X)$ on posets of subobjects has a left adjoint $\flat \dashv \sharp$:

$(U \hookrightarrow \sharp V) \Leftrightarrow (\flat U \hookrightarrow V) \,.$

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

###### Remark

We use the notation “$\flat$” and “$\sharp$” 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 “$\flat$” as “flat”.

###### Observation

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

(…)

###### Proposition

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

This appears as (AwodeyBirkedal, lemma 2.3).

###### Proof

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

$\sharp : \mathcal{E} \stackrel{\Gamma}{\to} Sh_j(\mathcal{E}) \stackrel{coDisc}{\hookrightarrow} \mathcal{E} \,,$

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

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

$\flat : \mathcal{E} \stackrel{\overset{Disc}{\hookleftarrow}}{\underset{\Gamma} {\to}} Sh_j(\mathcal{E}) \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}} \mathcal{E} : \sharp \,.$

Therefore the $(Disc \dashv \Gamma)$-counit provides morphisms

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

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

$\sharp (\flat X \to X) = (coDisc \Gamma Disc \Gamma \to coDisc \Gamma X)$

is an isomorphism.

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

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

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

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

is indeed an isomorphism.

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

###### Definition

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

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

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

###### Definition

An object $X \in \mathcal{E}$ is called o-discrete if $\flat X \simeq X$.

###### Lemma

Every discrete object is o-discrete.

### The axioms

###### Definition

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

• Axiom 1. $j$ is essential.

• Axiom 2 a. There is an object $G \in \mathcal{E}$ 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 $G'$ and an epimorphism $G' \to \flat G$.

• 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 $\mathcal{E} \to Sh_j(\mathcal{E}) \simeq D_j(\mathcal{E})$.

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

###### Proof

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

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

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

$Const(S) \colon C \to * \stackrel{S}{\to} Set \,.$

Notice that in the presence of a terminal object in $C$, $Const$ is a full and faithful functor: a natural transformation $Const(S_1) \to Const(S_2)$ has components

$\array{ S_1 = & Const(S_1)(*) &\stackrel{f}{\to}& Const(S_2)(*) & = S_2 \\ & \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ & Const(S_1)(U) &\to& Const(S_2)(U) }$

where the vertical morphisms are $Const(U \to *)$, 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 $f \colon S_1 \to S_2$.

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

$(\underset{\rightarrow}{\lim} \vdash Const \vdash \underset{\leftarrow}{\lim})$

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

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

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

\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 $[C^{op}, Set]$ 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 $CoDisc$.

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

\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 $(Const \vdash \Gamma \vdash CoDisc)$ where $Const$ is a full and faithful functor. By the discussion at fully faithful adjoint triples it follows then that also $CoDisc$ 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 $Sh(X)$ is local (over Set) iff $X$ has a focal point $x$, i.e. a point whose only neighborhood is the whole space. In this case, the extra right adjoint $f^! : Set \to Sh(X)$ to the global sections functor $f_* : Sh(X) \to Set$ is given by the functor which computes the stalk at $x$. This can also be given without reference to $x$, by the formula

$\Gamma(U, f^!(M)) = Hom(\{\star|U=X\}, M) \cong \begin{cases} M, & U = X, \\ \{\star\}, & U \neq X, \end{cases}$

for sets $M$ and open subsets $U \subseteq X$.

### Sheaves on a local site

For $C$ a local site, the category of sheaves $Sh(C)$ is a local topos over $Set$.

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

$Codisc : Set \to Sh(CartSp)$

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

### Localization

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

Note that if $E$ is a local topos with global sections geometric morphism $e^*\dashv e_*$, then the adjunction $e_*\dashv e^!$ is also a geometric morphism $Set\to E$. In this way we have a functor $LocTopos \to PTopos$, 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 $[C^{op},Set]$ at the point induced by $U\colon 1\to C$ can be identified with the presheaf topos $[(C/U)^{op},Set]$ over the over category of $C$ over $U$. By the general properties of over toposes, this is equivalently the over-topos $PSh(C)/U$ (where $U$ is regarded in $PSh(C)$ by the Yoneda embedding).

###### Proposition

If $X=Spec(A)$ 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 $Sh(X)$ at $P\colon 1\to X$ can be identified with $Sh(Spec(A_P))$, 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 $\mathcal{E}$ a Grothendieck topos and $X \in \mathcal{E}$ an object, the over topos $\mathcal{E}/X$ is local if $X$ is a tiny object (atomic object).

###### Proof

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

$\Gamma : (A \stackrel{f}{\to} X) \mapsto Hom_{\mathcal{E}/X}(Id_X, f) \,.$

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

$\array{ Hom_{\mathcal{E}/X}(Id_X, (A \to X)) &\to& Hom_{\mathcal{E}}(X,A) \\ \downarrow && \downarrow^{f_*} \\ * &\stackrel{Id_x}{\to}& Hom_{\mathcal{E}}(X,X) } \,.$

Moreover, overserve that colimits in the over category are computed in $\mathcal{E}$.

${\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X) \simeq ({\lim_\to}_i A_i) \to X \,.$

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

$Hom_{\mathcal{E}}(X, {\lim}_i A_i) \simeq {\lim_\to}_i Hom_{\mathcal{E}}(X, A_i) \,,$

Inserting all this into the above pullback gives the pullback

$\array{ Hom_{\mathcal{E}/X}(Id_X, {\lim_\to}_i (A_i \to X)) &\to& {\lim_\to}_i Hom_{\mathcal{E}}(X, A_i) \\ \downarrow && \downarrow^{f_*} \\ * &\stackrel{Id_x}{\to}& Hom_{\mathcal{E}}(X,X) } \,.$

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

$\Gamma({\lim_\to}_i (A_i \to X))) \simeq Hom_{\mathcal{E}/X}(Id_X, {\lim_\to}_i (A_i \to X)) \simeq {\lim_\to}_i Hom_{\mathcal{E}/X}(Id_X,(A_i \to X)) \simeq {\lim_\to}_i \Gamma(A_i \to X) \,.$

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

###### Remark

As a special case this reproduces the above statement that slices $PSh(C)/j(U)$ 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 $Spec A$ be the prime spectrum of $A$, and let $\mathcal{Z}_A$ 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 $D (a) = \{ \mathfrak{p} \in Spec A : a \notin \mathfrak{p} \}$, and these open subsets constitute a basis for the topology on $Spec A$. The full subcategory of the frame of open subsets of $Spec A$ 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 $\mathcal{Z}_A \to Sh (Spec A)$.

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 May 28, 2014 03:59:58 by Urs Schreiber (31.160.36.246)