nLab
exponentiable topos

Contents

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} the category 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}).

  • By setting =𝒮\mathcal{F}=\mathcal{S} , the base topos, one sees from Hom(𝒮,𝒢 )=Hom(𝒮×,𝒢)=Hom(,𝒢)Hom(\mathcal{S},\mathcal{G}^\mathcal{E})=Hom(\mathcal{S}\times\mathcal{E},\mathcal{G})=Hom(\mathcal{E},\mathcal{G}) that 𝒢 \mathcal{G}^\mathcal{E} is indeed a “mapping space” whose points are the geometric morphisms 𝒢\mathcal{E}\to\mathcal{G}.

  • The concept generalizes to higher topos theory (cf. Anel 2015, 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).

Exponentiability is a local property:

Proposition
  1. If a Grothendieck topos \mathcal{E} is exponentiable so is /X\mathcal{E}/X for any object XX\in\mathcal{E}.
  2. If /X\mathcal{E}/X is exponentiable and X1X\to 1 is an epimorphism, then \mathcal{E} is exponentiable as well.

This occurs as lemma 4.2 in Johnstone-Joyal (1982, p.281).

Continuity also leaves a lattice-theoretic trace:

Proposition

If a Grothendieck topos \mathcal{E} is exponentiable then the lattice of subobjects of any object XX\in\mathcal{E} is continuous.

This occurs as lemma 5.1 in Johnstone-Joyal (1982, p.287).

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!

Compare also the remarks of Anel (2015) on the \infty-case.

Examples

  • Let 00 be the initial topos that classifies the inconsistent theory 𝕋 1 \mathbb{T}^\emptyset_1 over the empty signature. Then from 𝒮[𝕆] 0=𝒮\mathcal{S}[\mathbb{O}]^0=\mathcal{S} follows that the dual of the inconsistent theory is 𝕋 1 *=𝕋 \mathbb{T}_1^{\emptyset\ast}=\mathbb{T}_{\emptyset}^{\emptyset} with 𝕋 \mathbb{T}_{\emptyset}^{\emptyset} the empty theory (over the empty signature).

  • From 𝒮[𝕆] 𝒮=𝒮[𝕆]\mathcal{S}[\mathbb{O}]^\mathcal{S}=\mathcal{S}[\mathbb{O}] follows in turn that the double dual of the inconsistent theory, or, in other words, the dual of the empty theory, is 𝕋 1 **=𝕋 *=𝕆\mathbb{T}_1^{\emptyset\ast\ast}=\mathbb{T}^{\emptyset\ast}_\emptyset=\mathbb{O} with 𝕆\mathbb{O} the theory of objects. (More generally, the dual theory of any dualizable theory is itself dualizable.)

  • Let 𝒮 2\mathcal{S}^2 be the Sierpinski topos that classifies subterminal objects or the theory of completely prime filters of the frame of opens of the Sierpinski space. Since ×𝒮 2 2\mathcal{E}\times \mathcal{S}^2\cong \mathcal{E}^2 it follows that

    2=Hom( 2,𝒮[𝕆])=Hom(×𝒮 2,𝒮[𝕆])=Hom(,𝒮[𝕆] 𝒮 2)=Hom(,𝒮[𝕆 2]),\mathcal{E}^2=Hom(\mathcal{E}^2,\mathcal{S}[\mathbb{O}])=Hom(\mathcal{E}\times\mathcal{S}^2,\mathcal{S}[\mathbb{O}])= Hom(\mathcal{E},\mathcal{S}[\mathbb{O}]^{\mathcal{S}^2})=Hom(\mathcal{E},\mathcal{S}[\mathbb{O}^2])\quad ,

    i.e. the dual theory is the theory of morphisms 𝕆 2\mathbb{O}^2.

The theory of sheaves on a coherent site

Recall, that given a small site (𝒞,J)(\mathcal{C},J) there is a geometric theory 𝔽 J\mathbb{F}_J called the theory of J-continuous flat functors that (albeit uneconomically) axiomaticizes the theory classified by Sh(𝒞,J)Sh(\mathcal{C},J) using the objects and morphisms in 𝒞\mathcal{C} for its signature. When Sh(𝒞,J)Sh(\mathcal{C},J) is exponentiable this theory is dualizable, and, in the case the site (𝒞,J c)(\mathcal{C},J_c) is coherent i. e. 𝒞\mathcal{C} has finite limits and J cJ_c is generated by finite covering families, the following theory 𝕊\mathbb{S} called the theory of sheaves on (𝒞,J c)(\mathcal{C},J_{c}) axiomatizes the dual theory 𝔽 J c *\mathbb{F}_{J_{c}}^\ast (in an equally uneconomic way):

The signature of 𝕊\mathbb{S} has a type symbol AA' for every object A𝒞A\in\mathcal{C} and a function symbol f:BAf':B'\to A' for every morphism f:BAf:B\to A in 𝒞 op\mathcal{C}^{op}. The axioms are given by the following schemata

  • xi(x)=x\top\vdash_x i'(x)=x for all identity morphisms ii.

  • xf(x)=h(g(x))\top\vdash_x f'(x)=h'(g'(x)) for all composable triples f=ghf=g\circ h in 𝒞\mathcal{C}.

  • i=1 n(α i(x)=a i(x)) x,xx=x\bigwedge_{i=1}^{n} (\alpha_i'(x)=a_i'(x'))\vdash_{x,x'} x=x' for each generating finite J cJ_{c}-covering family (Y iα iX|i=1,,n)(Y_i\overset{\alpha_i}{\to} X|i=1,\dots, n).

  • i,j(β ij(y i)=γ ij(y j)) y i,y jx( i=1 nα i(x)=y i)\bigwedge_{i,j}(\beta_{ij}'(y_i)=\gamma_{ij}'(y_j))\vdash_{y_i,y_j}\exists x\big ( \bigwedge_{i=1}^{n}\alpha_i'(x)=y_i\big ) for each generating finite J cJ_{c}-covering family (Y iα iX|i=1,,n)(Y_i\overset{\alpha_i}{\to} X|i=1,\dots, n) and pullback

    Z ij β ij Y i γ ij α i Y j α j X\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \array{ Z_{ij} &\overset{\beta_{ij}}{\longrightarrow}& Y_i \\ {}^{\mathllap{\gamma_{ij}}} \downarrow & & \downarrow^{\mathrlap{\alpha_i}} \\ Y_j &\underset{\alpha_j}{\longrightarrow}& X } .\qquad \qquad.

The models of 𝕊\mathbb{S} in a Grothendieck topos \mathcal{F} are “sheaves of \mathcal{F}-objects” on (𝒞,J c)(\mathcal{C},J_{c}) i.e. functors 𝒞 op\mathcal{C}^{op}\to\mathcal{F} satisfying the diagrammatic form of the sheaf axioms but

Mod 𝕊()=×Sh(𝒞,J c)=Hom(×Sh(𝒞,J c),𝒮[𝕆])=Hom(,𝒮[𝕆] Sh(𝒞,J c))Mod_\mathbb{S}(\mathcal{F})=\mathcal{F}\times Sh(\mathcal{C},J_{c})=Hom(\mathcal{F}\times Sh(\mathcal{C},J_{c}), \mathcal{S}[\mathbb{O}])=Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})})

i.e. 𝒮[𝕆] Sh(𝒞,J c)\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})} is the classifying topos for 𝕊\mathbb{S}. (For the details cf. Johnstone 1977, p.248f).

In particular, since 𝕊\mathbb{S} is a coherent theory this implies that 𝒮[𝕆] Sh(𝒞,J c)\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})} is also a coherent topos. From

Hom(𝒮,𝒮[𝕆] Sh(𝒞,J c))=Mod 𝕊(𝒮)=𝒮×Sh(𝒞,J c)=Sh(𝒞,J c)Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})})=Mod_\mathbb{S}(\mathcal{S})=\mathcal{S}\times {Sh(\mathcal{C},J_{c})}=Sh(\mathcal{C},J_c)

it follows then that every coherent topos Sh(𝒞,J c)Sh(\mathcal{C},J_c) is the category of points of a coherent topos.

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, Toposes are commutative rings , talk at Topos à l’IHES, IHES Paris, Nov. 25-27, 2015 (video recording, at min 52:47 ff.).

  • 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, Topos Theory , Academic Press New York 1977. (Also available as Dover reprint Mineola 2014; pp.209-210, 248-249)

  • Peter Johnstone, Sketches of an Elephant vols. 1&2 , CUP 2002. (sections B4.3 pp.432-438, C4.4 pp.745-754)

  • Peter Johnstone, André Joyal, Continuous categories and exponentiable toposes , JPAA 25 (1982) pp.255-296.

  • 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)

  • Myles Tierney, Forcing Topologies and Classifying Toposes , pp.211-219 in Heller, Tierney (eds.), Algebra, Topology and Category Theory , Academic Press New York 1976. (p.211, 218)

Last revised on October 2, 2020 at 17:52:22. See the history of this page for a list of all contributions to it.