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Definition

By ToposTopos (or ToposesToposes) is denoted the category of toposes. Usually this means:

This is naturally a 2-category, where

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

  • the (non-full) sub-2-category of Cat opCat^{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 coCat^{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=GrToposesShToposes = GrToposes of sheaf toposes (i.e. Grothendieck toposes).

  • Note that in some literature this 2-category is denoted merely TopTop, 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 LogLog or LogToposLogTopos.

Properties

From topological spaces to toposes

The operation of forming categories of sheaves

Sh():TopShToposes Sh(-) : Top \to ShToposes

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

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

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

Strictly speaking, this functor is not an embedding if we consider TopTop as a 1-category and ToposesToposes 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 TopTop 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 SobTopSobTop of sober spaces. This embedding can also be extended from SobTopSobTop to the entire category of locales (which can be viewed as “Grothendieck 0-toposes”).

From toposes to higher toposes

There are similar full embeddings ShToposSh2ToposShTopos \hookrightarrow Sh 2 Topos and ShToposSh(n,1)ToposShTopos \hookrightarrow Sh(n,1)Topos of sheaf (1-)toposes into 2-sheaf 2-toposes and sheaf (n,1)-toposes for 2n2\le n\le \infty.

From locally presentable categories to toposes

There is a canonical forgetful functor U:Topos 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 ToposTopos is not all that well-endowed with limits, but its slice categories are finitely complete as 2-categories, and ShToposShTopos is closed under finite limits in Topos/SetTopos/Set. In particular, the terminal object in ShToposesShToposes is the topos Set Sh(*)\simeq Sh(*).

Colimits

The supply with colimits is better:

Proposition

All small (indexed) 2-colimits in ShToposShTopos 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

p 2 2 p 1 f 2 1 f 1 \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 ToposTopos such that

then the diagram of inverse image functors

p 2 * 2 p 1 * f 2 * 1 f 1 * \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 ToposTopos is an extensive category. Same for toposes bounded over a base.

This is in (BungeLack, proposition 4.3).

Pullbacks

Proposition

Let

𝒳 (g *g *) 𝒴 (f *f *) 𝒵 \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 ToposTopos is computed, roughly, by the pushout of their sites of definition.

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

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

such that

( 𝒳 (g *g *) 𝒴 (f *f *) 𝒵)( Sh(𝒟) (Lan g()g) Sh(𝒟˜) (Lan f()f) Sh(𝒞)). \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

𝒟˜ 𝒞𝒟 f 𝒟 g g 𝒟˜ f 𝒞Cat 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 lexCat{}^{lex} \subset Cat of categories with finite limits.

Let moreover

Sh(𝒟˜ 𝒞𝒟)PSh(𝒟˜ 𝒞𝒟) 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()Lan_{f'}(-) of the coverings of 𝒟\mathcal{D} and the inverse image Lan g()Lan_{g'}(-) of the coverings of 𝒟˜\tilde \mathcal{D}.

Then

Sh(𝒟˜ 𝒞𝒟) 𝒳 (g *g *) 𝒴 (f *f *) 𝒵 \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 ToposTopos 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 ToposTopos 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 ToposTopos).

The adjunction between toposes and locally presentable categories is discussed in

category: category

Revised on December 31, 2013 20:58:06 by Tim Porter (2.26.35.109)