(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An exponentiable topos is a generalization of the notion of an exponentiable locale and can be viewed as a (topological) “space” $X$ that behaves well with respect to the construction of mapping spaces $Y^X$.
A Grothendieck topos $\mathcal{E}$ is called exponentiable (in the 2-category of Grothendieck toposes $GrTop$) if the 2-functor ${}_{-}\times\mathcal{E}$ has a right 2-adjoint $(_{-})^\mathcal{E}$.
More concretely, $\mathcal{E}$ is exponentiable if there exists a functor $(_{-})^\mathcal{E}$ such that for all toposes $\mathcal{F},\mathcal{G}$ $Hom(\mathcal{F}\times\mathcal{E},\mathcal{G})$ is (naturally) equivalent as a category to $Hom(\mathcal{F},\mathcal{G}^\mathcal{E})$.
The concept generalizes to higher topos theory (cf. Anel-Lejay 2018, Lurie 2018).
Interestingly, in the category of locales exponentiability of a locale $X$ hinges on the existence of the single exponential $S^X$ where $S$ is the Sierpinski space: $Y^X$ exists for all $Y$ iff $S^X$ exists.
In $GrTop$ the object classifier $\mathcal{S}[\mathbb{O}]$ takes over the role of the Sierpinski space and we have the following
A Grothendieck topos $\mathcal{E}$ is exponentiable iff the exponential $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ exists.
This result is due to Johnstone-Joyal (1982, p.282) and occurs as theorem 4.3.1 of Johnstone (2002, vol.1 p.433).
The following theorem pursues this analogy and generalizes a result of Martin Hyland on locales (1981).
A Grothendieck topos $\mathcal{E}$ is an exponentiable object in the 2-category of Grothendieck toposes and geometric morphisms iff $\mathcal{E}$ is a continuous category.
This result is due to Johnstone-Joyal (1982) and occurs as theorem 4.4.5 of Johnstone (2002, p.748).
Since locally finitely presentable categories are continuous and coherent toposes are locally finitely presentable (cf. Johnstone (2002, p.915)) it follows that coherent toposes are exponentiable. This can be viewed as an avatar of the fact that (locally) compact topological spaces behave well with respect to mapping spaces.
By the same reasoning all functor categories $Set^{\mathcal{C}}$ for $\mathcal{C}$ a small category are exponentiable since they are locally finitely presentable. This includes in particular all presheaf toposes on small categories.
It is worthwhile to muse a bit about $\mathcal{S}[\mathbb{O}]^\mathcal{E}$:
First of at all, it is injective since $\mathcal{S}[\mathbb{O}]$ is injective and $(_-)^\mathcal{E}$ preserves inclusions.
This and the various universal properties of the toposes involved imply that an exponentiable topos
is (up to equivalence) the category of points of an injective topos.
Now consider an arbitrary topos $\mathcal{E}$ classifying a geometric theory $\mathbb{T}$.
Then from the functorial perspective on logic the 2-functor $\mathbb{T}^*:GrTop^{op}\to CAT$ assigning to $\mathcal{F}$ the category of models $\mathbb{T}^*(\mathcal{F}):=\mathcal{F}\times\mathcal{E}=\mathcal{F}\times\mathcal{S}[\mathbb{T}]$ is called the dual theory of $\mathbb{T}$.
It need not be geometric i.e. have a classifying topos but when it is, $\mathbb{T}$ being called dualizable in that case, the following
shows that $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ has precisely the properties required for $\mathcal{S}[\mathbb{T}^*]$.
In other words, $\mathbb{T}^*$ is geometric (and classified by $\mathcal{S}[\mathbb{O}]^\mathcal{E}$) precisely iff $\mathcal{E}=\mathcal{S}[\mathbb{T}]$ is exponentiable!
The notion of a tiny object in a cartesian closed category suggests the following
An exponentiable Grothendieck topos $\mathcal{E}$ is called tiny (or infinitesimal) if the 2-functor $(_{-})^\mathcal{E}$ has a right 2-adjoint $(_{-})_\mathcal{E}$.
Mathieu Anel, Damien Lejay, Exponentiable Higher Toposes , arXiv:1802.10425 (2018). (abstract)
Andreas Blass, The interaction of category theory and set theory , Cont. Math. 30 (1984) pp.5-29. (draft)
Martin Hyland, Function spaces in the category of locales , Springer LNM 871 (1981) pp.264-281.
Peter Johnstone, André Joyal, Continuous categories and exponentiable toposes , JPAA 25 (1982) pp.255-296.
Peter Johnstone, Sketches of an Elephant vols. 1&2 , CUP 2002. (sections B4.3 pp.432-438, C4.4 pp.745-754)
Jacob Lurie, Spectral Algebraic Geometry , ms. Harvard University 2018. (section 21.1.6)
Susan Niefield, Exponentiable Morphisms: posets, spaces, locales and Grothendieck toposes , TAC 8 (2001) pp.16-32. (abstract)
Last revised on March 22, 2018 at 08:11:16. See the history of this page for a list of all contributions to it.