Contents

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

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

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

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

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 $(LConst \dashv \Gamma \dashv CoDisc)$

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

(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.)

Note also that in an infinitary-extensive category an object is connected as soon as $\hom(X,-)$ preserves binary coproducts (see connected object). Moreover, a coproduct-preserving functor between toposes preserves coequalizers as soon as it preserves epimorphisms, since any coequalizer can be constructed as the quotient of an equivalence relation generated using images and countable coproducts and quotients of equivalence relations are effective. Thus, we can say that a Grothendieck topos is local iff $\hom(1,-)$ preserves binary coproducts and epis. Moreover, a cospan $A \to C \leftarrow B$ in a topos is a coproduct diagram iff $A$ and $B$ are disjoint monos whose union is all of $C$; thus $\hom(1,-)$ preserves binary coproducts as soon as it preserves the initial object and binary unions. This leads to the following equivalent form of “locality” that makes sense even for elementary toposes:

###### Definition

An elementary topos $E$ is local if the terminal object $1$ is

• nonempty: $1\ncong 0$,

• connected: $1 = p \vee q$ implies $1 = p$ or $1 = q$, and

• projective: every epi $U \to 1$ admits a section $t: 1 \to U$.

###### Remark

Some authors have instead used the term “local” to refer just to the condition that $1$ is connected; note this is equivalent to $Sub_E(1)$ being a local rig. In his thesis, Steve Awodey upgraded this, saying that a topos is hyperlocal if both connectivity and projectivity are satisfied. As argued above, for a Grothendieck topos this is slightly weaker than locality, although the difference is only in the inclusion of the trivial topos (for if $1\cong 0$ then $E\simeq 1$). Note that a local elementary topos, as defined above, is constructively well-pointed if and only if $1$ is additionally a generator.

###### Example

The free topos is a local elementary topos.

## 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

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

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

Notions of local topos, with a view to logical completeness theorems, appear in Steve Awodey’s thesis:

• Steve Awodey, Logic in topoi: functorial semantics for higher-order logic, Thesis, University of Chicago (1997).