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Definition

By $Topos$ (or $Toposes$) 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, $Toposes$ 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.

Related 2-categories

• 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

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

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:

This appears as (Moerdijk, theorem 2.5)

This appears as theorem 5.1 in (BungeLack)

This is in (BungeLack, proposition 4.3).

Pullbacks

This appears for instance as (Lurie, prop. 6.3.4.6).

Free loop spaces

The free loop space object of a topos in Topos is called the isotropy group of a topos.

Related concepts

• Topos

• (∞,1)Topos

References

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

• 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 $Topos$).

The adjunction between toposes and locally presentable categories is discussed in

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