nLab
exponentiable topos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

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” XX that behaves well with respect to the construction of mapping spaces Y XY^X.

Definition

Definition

A Grothendieck topos \mathcal{E} is called exponentiable (in the 2-category of Grothendieck toposes GrTopGrTop) 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 ,𝒢\mathcal{F},\mathcal{G} Hom(×,𝒢)Hom(\mathcal{F}\times\mathcal{E},\mathcal{G}) is (naturally) equivalent as a category to Hom(,𝒢 )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 XX hinges on the existence of the single exponential S XS^X where SS is the Sierpinski space: Y XY^X exists for all YY iff S XS^X exists.

In GrTopGrTop the object classifier 𝒮[𝕆]\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 𝒮[𝕆] \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).

Examples

Remarks on duality

It is worthwhile to muse a bit about 𝒮[𝕆] \mathcal{S}[\mathbb{O}]^\mathcal{E}:

First of at all, it is injective since 𝒮[𝕆]\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

Hom(,𝒮[𝕆])Hom(𝒮×,𝒮[𝕆])Hom(𝒮,𝒮[𝕆] )\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 𝕋\mathbb{T}.

Then from the functorial perspective on logic the 2-functor 𝕋 *:GrTop opCAT\mathbb{T}^*:GrTop^{op}\to CAT assigning to \mathcal{F} the category of models 𝕋 *():=×=×𝒮[𝕋]\mathbb{T}^*(\mathcal{F}):=\mathcal{F}\times\mathcal{E}=\mathcal{F}\times\mathcal{S}[\mathbb{T}] is called the dual theory of 𝕋\mathbb{T}.

It need not be geometric i.e. have a classifying topos but when it is, 𝕋\mathbb{T} being called dualizable in that case, the following

𝕋 *()=×Hom(×,𝒮[𝕆])Hom(,𝒮[𝕆] )\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 𝒮[𝕆] \mathcal{S}[\mathbb{O}]^\mathcal{E} has precisely the properties required for 𝒮[𝕋 *]\mathcal{S}[\mathbb{T}^*].

In other words, 𝕋 *\mathbb{T}^* is geometric (and classified by 𝒮[𝕆] \mathcal{S}[\mathbb{O}]^\mathcal{E}) precisely iff =𝒮[𝕋]\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}.

References

  • 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.