nLab
solid topos

Contents

under construction

Context

Cohesive Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Contents

Idea

The concept of solid topos is meant to be a characterization of gros toposes which further refines that of elastic toposes and cohesive toposes. The idea is that we may consistently regard the objects of such toposes as generalized spaces for some flavor of geometry, and that the axioms on the topos determine aspects of the geometric nature of these generalized spaces, as follows:

A\phantom{A}gros toposA\phantom{A}A\phantom{A}generalized spaces obey…A\phantom{A}
A\phantom{A}cohesionA\phantom{A}principles of differential topologyA\phantom{A}
A\phantom{A}elasticityA\phantom{A}principles of differential geometryA\phantom{A}
A\phantom{A}solidityA\phantom{A}A\phantom{A}principles of supergeometryA\phantom{A}

Definition

Definition

(solid topos)

Let H bos\mathbf{H}_{bos} be an elastic topos (this Def.) over a cohesive topos H red\mathbf{H}_{red} (this Def.). Then a solid topos or super-differentially cohesive topos over H bos\mathbf{H}_{bos} is a sheaf topos H\mathbf{H}, which is

  1. a cohesive topos over Set (this Def.),

  2. an elastic topos over H red\mathbf{H}_{red} (this Def.).

  3. equipped with a quadruple of adjoint functors (this Def.) to H bos\mathbf{H}_{bos} of the form

    H bosAevenA AAι supAA AAΠ supAA AADisc supAAH \mathbf{H}_{bos} \array{ \overset{\phantom{A} even \phantom{A} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} } \mathbf{H}

    hence with ι sup\iota_{sup} and Disc supDisc_{sup} being fully faithful functors.

Lemma

(progression of (co-)reflective subcategories of solid topos)

Let H\mathbf{H} be a solid topos (Def. ) over an elastic topos H red\mathbf{H}_{red}:

SetAΠ redA ADisc redA AΓ redA AcoDisc redAH redAAι infAA AAΠ infAA AADisc infAA AAΓ infAA A AAH bosAAevenAA AAι supAA AAΠ supAA AADisc supAA AAΓ supAA A AA AAH Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} } \mathbf{H}_{bos} \array{ \overset{\phantom{AA} even \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{sup} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} \\ \phantom{A \atop A} } \mathbf{H}

Then these adjoint functors arrange and decompose as shown in the following diagram:

Here the composite adjoint quadruple

SetΠΠ redΠ infΠ sup Disc=Disc supDisc infDisc red Γ=Γ supΓ infΓ red AAcoDiscAAH Set \array{ \overset{\Pi \simeq \Pi_{red}\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc = Disc_{sup} Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{\Gamma = \Gamma_{sup} \Gamma_{inf} \Gamma_{red} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}

exhibits the cohesion of H\mathbf{H} over Set, and the composite adjoint quadruple

H redι supι inf Π infΠ sup Disc infDisc red Γ supH \mathbf{H}_{red} \array{ \overset{\iota_{sup} \iota_{inf}}{\hookrightarrow} \\ \overset{\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{ \Gamma_{sup} }{\longleftarrow} } \mathbf{H}

exhibits the elasticity of H\mathbf{H} over H red\mathbf{H}_{red}.

Proof

As in the proof of this Prop., this is immediate by the essential uniqueness of adjoints (this Prop.) and of the global section-geometric morphism (this Example).

Definition

(adjoint modalities on solid topos)

Given a solid topos H\mathbf{H} over H bos\mathbf{H}_{bos} (Def. ), composition of the functors in Lemma yields, via this Prop., the following adjoint modalities (this Def.)

Rh:Hι supeven ι supΠ sup RhDisc supΠ supH. \rightrightarrows \;\dashv\; \rightsquigarrow \;\dashv\; Rh \;\;\colon\;\; \mathbf{H} \array{ \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even }{\longleftarrow} \\ \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} }{\longrightarrow} \\ \overset{ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} }{\longleftarrow} } \mathbf{H} \,.

Since ι sup\iota_{sup} and Disc supDisc_{sup} are fully faithful functors by assumption, these are (co-)modal operators (this Def.) on the cohesive topos, by this Prop..

We pronounce these as follows:

A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

and we refer to the corresponding modal objects (this Def.) as follows:

  • a \rightsquigarrow-comodal object

    Xϵ X X \overset{\rightsquigarrow}{X} \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow} X

    is called a bosonic object;

  • a RhRh-modal object

    Xη X RhRhX X \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow} Rh X

    is called a rheonomic object;

Proposition

(progression of adjoint modalities on solid topos)

Let H\mathbf{H} be a solid topos (Def. ) and consider the adjoint modalities which it inherits

  1. for being a cohesive topos, from this Def.,

  2. for being an elastic topos, from this Def.,

  3. for being a solid topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠʃ \;\coloneqq\; Disc \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι supι infΠ infΠ sup\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup} A\phantom{A}A\phantom{A} Disc supDisc infΠ infΠ sup\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup} A\phantom{A}A\phantom{A} &Disc supDisc infΓ infΓ sup \& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup} A\phantom{A}
A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from this Def.:

id id Rh & ʃ * \array{ id &\dashv& id \\ \vee && \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee && \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee && \vee \\ && && ʃ &\dashv& \flat &\dashv& \sharp \\ && && && \vee && \vee \\ && && && \emptyset &\dashv& \ast }

where we are displaying, for completeness, also the adjoint modalities at the bottom *\emptyset \dashv \ast and the top ididid \dashv id (this Example).

Proof

By this Prop., it just remains to show that for all objects XHX \in \mathbf{H}

  1. X\Im X is an RhRh-modal object, hence RhXXRh \Im X \simeq X,

  2. X\Re X is a bosonic object, hence XX\overset{\rightsquigarrow}{\Re X} \simeq \Re X.

The proof is directly analogous to the proof of that Prop., now using the decompositions from Lemma :

Rh =Disc supΠ supDisc supidDisc infΠ infΠ sup Disc supDisc infΠ infΠ sup = \begin{aligned} Rh \Im & = Disc_{sup} \underset{ \simeq id }{ \underbrace{ \Pi_{sup} \;\; Disc_{sup} } } Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & \simeq Disc_{sup} Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & = \Im \end{aligned}

and

=ι supΠ supι supidι infΠ infΠ sub ι supι infΠ infΠ sub \begin{aligned} \rightsquigarrow \Re & = \iota_{sup} \underset{\simeq id}{\underbrace{ \Pi_{sup} \;\; \iota_{sup}}} \iota_{inf} \Pi_{inf}\Pi_{sub} \\ & \simeq \iota_{sup} \iota_{inf} \Pi_{inf} \Pi_{sub} \\ & \simeq \Re \end{aligned}

Examples

Super formal smooth sets

The sheaf topos of super formal smooth sets is solid over that of formal smooth sets, which is elastic over that of smooth sets, which is cohesive over Set.

See this Prop at geometry of physics – supergeometry.

Last revised on July 31, 2018 at 07:31:52. See the history of this page for a list of all contributions to it.