nLab
Topos

Definition

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

• objects are toposes;

• morphisms are geometric morphisms of toposes.

This is naturally a 2-category, where

• 2-morphism are geometric transformations

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.

Related 2-categories

• 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

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

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

$$\mathrm{Sh}(f):\mathrm{Sh}(X)\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}\mathrm{Sh}(Y)$$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 $\mathrm{ShTopos}\hookrightarrow \mathrm{Sh}2\mathrm{Topos}$ and $\mathrm{ShTopos}\hookrightarrow \mathrm{Sh}(n,1)\mathrm{Topos}$ of sheaf (1-)toposes into 2-sheaf 2-toposes and sheaf (n,1)-toposes for $2\le n\le \mathrm{\infty }$.

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 $\simeq \mathrm{Sh}(*)$.

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

$$\begin{array}{ccc}ℱ& \stackrel{p_2}{\to }& {ℰ}_2\\ {}^{p_1}\downarrow & ⇙& {\downarrow }^{f_2}\\ {ℰ}_1& \underset{f_1}{\to }& ℰ\end{array}$$\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

• $f_1,f_2$ are both pseudomonic morphisms

• ${ℰ}_1\coprod {ℰ}_2\to ℰ$ is an effective epimorphism;

then the diagram of inverse image functors

$$\begin{array}{ccc}ℱ& \stackrel{p_2^{*}}{\leftarrow }& {ℰ}_2\\ {}^{p_1^{*}}\uparrow & ⇙& {\uparrow }^{f_2^{*}}\\ {ℰ}_1& \underset{f_1^{*}}{\leftarrow }& ℰ\end{array}$$\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

$$\begin{array}{ccc}& & 𝒳\\ & & {\downarrow }^{(g^{*}⊣g_{*})}\\ 𝒴& \stackrel{(f^{*}⊣f_{*})}{\to }& 𝒵\end{array}$$\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 $\widetilde{𝒟}$, $𝒟$, and $𝒞$ with finite limits and morphisms of sites

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

such that

$$\left(\begin{array}{ccc}& & 𝒳\\ & & {\downarrow }^{(g^{*}⊣g_{*})}\\ 𝒴& \stackrel{(f^{*}⊣f_{*})}{\to }& 𝒵\end{array}\right)\simeq \left(\begin{array}{ccc}& & \mathrm{Sh}(𝒟)\\ & & {\downarrow }^{({\mathrm{Lan}}_g⊣(-)\circ g)}\\ \mathrm{Sh}(\widetilde{𝒟})& \stackrel{({\mathrm{Lan}}_f⊣(-)\circ f)}{\to }& \mathrm{Sh}(𝒞)\end{array}\right).$$\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

$$\begin{array}{ccc}\widetilde{𝒟}\coprod _{𝒞}𝒟& \stackrel{f\prime }{\leftarrow }& 𝒟\\ {}^{g\prime }\uparrow & {⇙}_{\simeq }& {\uparrow }^g\\ \widetilde{𝒟}& \stackrel{f}{\leftarrow }& 𝒞\end{array}\in {\mathrm{Cat}}^{\mathrm{lex}}$$\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

$$\mathrm{Sh}(\widetilde{𝒟}\coprod _{𝒞}𝒟)\hookrightarrow \mathrm{PSh}(\widetilde{𝒟}\coprod _{𝒞}𝒟)$$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 ${\mathrm{Lan}}_{f\prime }(-)$ of the coverings of $𝒟$ and the inverse image ${\mathrm{Lan}}_{g\prime }(-)$ of the coverings of $\widetilde{𝒟}$.

Then

$$\begin{array}{ccc}\mathrm{Sh}(\widetilde{𝒟}\coprod _{𝒞}𝒟)& \to & 𝒳\\ \downarrow & & {\downarrow }^{(g^{*}⊣g_{*})}\\ 𝒴& \stackrel{(f^{*}⊣f_{*})}{\to }& 𝒵\end{array}$$\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.

Related concepts

• Topos

• (∞,1)Topos

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

• Marta Bunge, Steve Lack, van Kampen theorem for toposes (ps)

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

• Jacob Lurie, Higher Topos Theory .

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