Cohomology and homotopy
In higher category theory
By (or ) is denoted the category of toposes. Usually this means:
This is naturally a 2-category, where
That is, a 2-morphism is a natural transformation (which is, by mate calculus, equivalent to a natural transformation between direct images). Thus, is equivalent to both of
- the (non-full) sub-2-category of on categories that are toposes and morphisms that are the inverse image parts of geometric morphisms, and
- the (non-full) sub-2-category of on categories that are toposes and morphisms that are the direct image parts of geometric morphisms.
There is also the sub-2-category of sheaf toposes (i.e. Grothendieck toposes).
Note that in some literature this 2-category is denoted merely , 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 or .
From topological spaces to toposes
The operation of forming categories of sheaves
embeds topological spaces into toposes. For a continuous map we have that is the geometric morphism
with the direct image and the inverse image.
Strictly speaking, this functor is not an embedding if we consider as a 1-category and 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 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 of sober spaces. This embedding can also be extended from to the entire category of locales (which can be viewed as “Grothendieck 0-toposes”).
From toposes to higher toposes
There are similar full embeddings and of sheaf (1-)toposes into 2-sheaf 2-toposes and sheaf (n,1)-toposes for .
From locally presentable categories to toposes
There is a canonical forgetful functor 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 is not all that well-endowed with limits, but its slice categories are finitely complete as 2-categories, and is closed under finite limits in . In particular, the terminal object in is the topos Set .
The supply with colimits is better:
This appears as (Moerdijk, theorem 2.5)
be a 2-pullback in such that
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 is an extensive category. Same for toposes bounded over a base.
This is in (BungeLack, proposition 4.3).
be a diagram of toposes. Then its pullback in the (2,1)-category version of is computed, roughly, by the pushout of their sites of definition.
More in detail: there exist sites , , and with finite limits and morphisms of sites
be the pushout of the underlying categories in the full subcategory Cat of categories with finite limits.
be the reflective subcategory obtained by localization at the class of morphisms generated by the inverse image of the coverings of and the inverse image of the coverings of .
is a pullback square.
This appears for instance as (Lurie, prop. 188.8.131.52).
The characterization of colimits in 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 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 ).
The adjunction between toposes and locally presentable categories is discussed in