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.


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(*).


The supply with colimits is better:


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)



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)


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

This is in (BungeLack, proposition 4.3).




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}.


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.


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.


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 (