Could not include topos theory - contents
By $Topos$ (or $Toposes$) is denoted the category of toposes. Usually this means:
morphisms are geometric morphisms of toposes.
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
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 $LogTopos$.
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”).
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 $2\le n\le \infty$.
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).
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 $\simeq Sh(*)$.
The supply with colimits is better:
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)
Let
be a 2-pullback in $Topos$ such that
$f_1, f_2$ are both pseudomonic morphisms
$\mathcal{E}_1 \coprod \mathcal{E}_2 \to \mathcal{E}$ is an effective epimorphism;
then the diagram of inverse image functors
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 $Topos$ is an extensive category. Same for toposes bounded over a base.
This is in (BungeLack, proposition 4.3).
Let
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
such that
Let then
be the pushout of the underlying categories in the full subcategory Cat${}^{lex} \subset Cat$ of categories with finite limits.
Let moreover
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
is a pullback square.
This appears for instance as (Lurie, prop. 6.3.4.6).
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.
Topos
The characterization of colimits in $Topos$ is in
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