topos theory

category theory

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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