topos theory

category theory

# Contents

## Definition

By $\mathrm{Topos}$ (or $\mathrm{Toposes}$) 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, $\mathrm{Toposes}$ is equivalent to both of

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

• Note that in some literature this 2-category is denoted merely $\mathrm{Top}$, 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 $\mathrm{Log}$ or $\mathrm{LogTopos}$.

## Properties

### From topological spaces to toposes

The operation of forming categories of sheaves

$\mathrm{Sh}\left(-\right):\mathrm{Top}\to \mathrm{ShToposes}$Sh(-) : Top \to ShToposes

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

$\mathrm{Sh}\left(f\right):\mathrm{Sh}\left(X\right)\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}\mathrm{Sh}\left(Y\right)$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 $\mathrm{Top}$ as a 1-category and $\mathrm{Toposes}$ 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 $\mathrm{Top}$ 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 $\mathrm{SobTop}$ of sober spaces. This embedding can also be extended from $\mathrm{SobTop}$ to the entire category of locales (which can be viewed as “Grothendieck 0-toposes”).

### From toposes to higher toposes

There are similar full embeddings $\mathrm{ShTopos}↪\mathrm{Sh}2\mathrm{Topos}$ and $\mathrm{ShTopos}↪\mathrm{Sh}\left(n,1\right)\mathrm{Topos}$ of sheaf (1-)toposes into 2-sheaf 2-toposes and sheaf (n,1)-toposes for $2\le n\le \infty$.

### From locally presentable categories to toposes

There is a canonical forgetful functor $U:\mathrm{Topos}\to$ 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 $\mathrm{Topos}$ is not all that well-endowed with limits, but its slice categories are finitely complete as 2-categories, and $\mathrm{ShTopos}$ is closed under finite limits in $\mathrm{Topos}/\mathrm{Set}$. In particular, the terminal object in $\mathrm{ShToposes}$ is the topos Set $\simeq \mathrm{Sh}\left(*\right)$.

#### Colimits

The supply with colimits is better:

###### Proposition

All small (indexed) 2-colimits in $\mathrm{ShTopos}$ 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

$\begin{array}{ccc}ℱ& \stackrel{{p}_{2}}{\to }& {ℰ}_{2}\\ {}^{{p}_{1}}↓& ⇙& {↓}^{{f}_{2}}\\ {ℰ}_{1}& \underset{{f}_{1}}{\to }& ℰ\end{array}$\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 $\mathrm{Topos}$ such that

then the diagram of inverse image functors

$\begin{array}{ccc}ℱ& \stackrel{{p}_{2}^{*}}{←}& {ℰ}_{2}\\ {}^{{p}_{1}^{*}}↑& ⇙& {↑}^{{f}_{2}^{*}}\\ {ℰ}_{1}& \underset{{f}_{1}^{*}}{←}& ℰ\end{array}$\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 $\mathrm{Topos}$ is an extensive category. Same for toposes bounded over a base.

This is in (BungeLack, proposition 4.3).

#### Pullbacks

###### Proposition

Let

$\begin{array}{ccc}& & 𝒳\\ & & {↓}^{\left({g}^{*}⊣{g}_{*}\right)}\\ 𝒴& \stackrel{\left({f}^{*}⊣{f}_{*}\right)}{\to }& 𝒵\end{array}$\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 $\mathrm{Topos}$ is computed, roughly, by the pushout of their sites of definition.

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

$\begin{array}{ccc}& & 𝒟\\ & & {↑}^{g}\\ \stackrel{˜}{𝒟}& \stackrel{f}{←}& 𝒞\end{array}$\array{ && \mathcal{D} \\ && \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} }

such that

$\left(\begin{array}{ccc}& & 𝒳\\ & & {↓}^{\left({g}^{*}⊣{g}_{*}\right)}\\ 𝒴& \stackrel{\left({f}^{*}⊣{f}_{*}\right)}{\to }& 𝒵\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\simeq \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\begin{array}{ccc}& & \mathrm{Sh}\left(𝒟\right)\\ & & {↓}^{\left({\mathrm{Lan}}_{g}⊣\left(-\right)\circ g\right)}\\ \mathrm{Sh}\left(\stackrel{˜}{𝒟}\right)& \stackrel{\left({\mathrm{Lan}}_{f}⊣\left(-\right)\circ f\right)}{\to }& \mathrm{Sh}\left(𝒞\right)\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟& \stackrel{f\prime }{←}& 𝒟\\ {}^{g\prime }↑& {⇙}_{\simeq }& {↑}^{g}\\ \stackrel{˜}{𝒟}& \stackrel{f}{←}& 𝒞\end{array}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\in {\mathrm{Cat}}^{\mathrm{lex}}$\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${}^{\mathrm{lex}}\subset \mathrm{Cat}$ of categories with finite limits.

Let moreover

$\mathrm{Sh}\left(\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟\right)↪\mathrm{PSh}\left(\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟\right)$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 ${\mathrm{Lan}}_{f\prime }\left(-\right)$ of the coverings of $𝒟$ and the inverse image ${\mathrm{Lan}}_{g\prime }\left(-\right)$ of the coverings of $\stackrel{˜}{𝒟}$.

Then

$\begin{array}{ccc}\mathrm{Sh}\left(\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟\right)& \to & 𝒳\\ ↓& & {↓}^{\left({g}^{*}⊣{g}_{*}\right)}\\ 𝒴& \stackrel{\left({f}^{*}⊣{f}_{*}\right)}{\to }& 𝒵\end{array}$\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.

## References

The characterization of colimits in $\mathrm{Topos}$ 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 $\mathrm{Topos}$ 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 $\mathrm{Topos}$).

The adjunction between toposes and locally presentable categories is discussed in

• Marta Bunge, Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, (1995)

category: category

Revised on May 8, 2012 09:39:03 by Urs Schreiber (89.204.137.28)