Contents

topos theory

category theory

# Contents

## Definition

By $Topos$ (or $Toposes$) is denoted the category of toposes. Usually this means:

This is naturally a 2-category, where

That is, a 2-morphism $f\to g$ is a natural transformation $f^* \to g^*$ (which is, by mate calculus, equivalent to a natural transformation $g_* \to f_*$ between direct images). Thus, $Toposes$ is equivalent to both of

• the (non-full) sub-2-category of $Cat^{op}$ on categories that are toposes and morphisms that are the inverse image parts of geometric morphisms, and
• the (non-full) sub-2-category of $Cat^{co}$ on categories that are toposes and morphisms that are the direct image parts of geometric morphisms.
• There is also the sub-2-category $ShToposes = GrToposes$ of sheaf toposes (i.e. Grothendieck toposes).

• Note that in some literature this 2-category is denoted merely $Top$, but that is also commonly used to denote the category of topological spaces.

• We obtain a very different 2-category of toposes if we take the morphisms to be logical functors; this 2-category is sometimes denoted $Log$ or $LogTopos$.

## Properties

### From topological spaces to toposes

The operation of forming categories of sheaves

$Sh(-) : Top \to ShToposes$

embeds topological spaces into toposes. For $f : X \to Y$ a continuous map we have that $Sh(f)$ is the geometric morphism

$Sh(f) : Sh(X) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} Sh(Y)$

with $f_*$ the direct image and $f^*$ the inverse image.

Strictly speaking, this functor is not an embedding if we consider $Top$ as a 1-category and $Toposes$ as a 2-category, since it is then not fully faithful in the 2-categorical sense—there can be nontrivial 2-cells between geometric morphisms between toposes of sheaves on topological spaces.

However, if we regard $Top$ as a (1,2)-category where the 2-cells are inequalities in the specialization ordering, then this functor does become a 2-categorically full embedding (i.e. an equivalence on hom-categories) if we restrict to the full subcategory $SobTop$ of sober spaces. This embedding can also be extended from $SobTop$ to the entire category of locales (which can be viewed as “Grothendieck 0-toposes”).

### From toposes to higher toposes

There are similar full embeddings $ShTopos \hookrightarrow Sh 2 Topos$ and $ShTopos \hookrightarrow Sh(n,1)Topos$ of sheaf (1-)toposes into 2-sheaf 2-toposes and sheaf (n,1)-toposes for $2\le n\le \infty$. Note that these embeddings are not the identity functor on underlying categories: a 1-topos is not itself an $n$-topos, instead we have to take $n$-sheaves on a suitable generating site for it.

### From locally presentable categories to toposes

There is a canonical forgetful functor $U : Topos \to$ Cat that lands, by definition, in the sub-2-category of locally presentable categories and functors which preserve all limits / are right adjoints.

This 2-functor has a right 2-adjoint (Bunge-Carboni).

### Limits and colimits

The 2-category $Topos$ is not all that well-endowed with limits, but its slice categories are finitely complete as 2-categories, and $ShTopos$ is closed under finite limits in $Topos/Set$. In particular, the terminal object in $ShToposes$ is the topos Set $\simeq Sh(*)$.

#### Colimits

The supply with colimits is better:

###### Proposition

All small (indexed) 2-colimits in $ShTopos$ exists and are computed as (indexed) 2-limits in Cat of the underlying inverse image functors.

This appears as (Moerdijk, theorem 2.5)

###### Proposition

Let

$\array{ \mathcal{F} &\stackrel{p_2}{\to}& \mathcal{E}_2 \\ {}^{\mathllap{p_1}}\downarrow &\swArrow& \downarrow^{\mathrlap{f_2}} \\ \mathcal{E}_1 &\underset{f_1}{\to}& \mathcal{E} }$

be a 2-pullback in $Topos$ such that

then the diagram of inverse image functors

$\array{ \mathcal{F} &\stackrel{p_2^*}{\leftarrow}& \mathcal{E}_2 \\ {}^{\mathllap{p_1^*}}\uparrow &\swArrow& \uparrow^{\mathrlap{f_2^*}} \\ \mathcal{E}_1 &\underset{f_1^*}{\leftarrow}& \mathcal{E} }$

is a 2-pullback in Cat and so by the above the original square is also a 2-pushout.

This appears as theorem 5.1 in (BungeLack)

###### Proposition

The 2-category $Topos$ is an extensive category. Same for toposes bounded over a base.

This is in (BungeLack, proposition 4.3).

#### Pullbacks

###### Proposition

Let

$\array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }$

be a diagram of toposes. Then its pullback in the (2,1)-category version of $Topos$ is computed, roughly, by the pushout of their sites of definition.

More in detail: there exist sites $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with finite limits and morphisms of sites

$\array{ && \mathcal{D} \\ && \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} }$

such that

$\left( \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } \right) \,\,\, \simeq \,\,\, \left( \array{ && Sh(\mathcal{D}) \\ && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}} \\ Sh(\tilde \mathcal{D}) &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}& Sh(\mathcal{C}) } \right) \,.$

Let then

$\array{ \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D} \\ {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} } \,\,\,\,\, \in Cat^{lex}$

be the pushout of the underlying categories in the full subcategory Cat${}^{lex} \subset Cat$ of categories with finite limits.

Let moreover

$Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) \hookrightarrow PSh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})$

be the reflective subcategory obtained by localization at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the coverings of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.

Then

$\array{ Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) &\to& \mathcal{X} \\ \downarrow && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }$

is a pullback square.

This appears for instance as (Lurie, prop. 6.3.4.6).

###### Remark

For localic toposes this reduces to the statement of localic reflection: the pullback of toposes is given by the of the underlying locales which in turn is the pushout of the corresponding frames.

### Free loop spaces

The free loop space object of a topos in Topos is called the isotropy group of a topos.

The characterization of colimits in $Topos$ is in

• Ieke Moerdijk, The classifying topos of a continuous groupoid. I Transaction of the American mathematical society Volume 310, Number 2, (1988) (pdf)

The fact that $Topos$ is extensive is in

Limits and colimits of toposes are discussed in 6.3.2-6.3.4 of

There this is discussed for for (∞,1)-toposes, but the statements are verbatim true also for ordinary toposes (in the (2,1)-category version of $Topos$).

The adjunction between toposes and locally presentable categories is discussed in

category: category