# nLab exponentiable topos

Contents

topos theory

## Theorems

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

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 $Y^X$.

## Definition

###### Definition

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 $(_{-})^\mathcal{E}$.

## Remarks

• 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).

## Properties

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

###### Proposition

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).

###### Theorem

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).

## Remarks on duality

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

$\mathcal{E}\cong Hom(\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^\mathcal{E})$

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

$\mathbb{T}^*(\mathcal{F})=\mathcal{F}\times\mathcal{E}\cong Hom(\mathcal{F}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^\mathcal{E})$

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!

## Ramifications

The notion of a tiny object in a cartesian closed category suggests the following

###### Definition

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)