relative flat modality



Cohesive \infty-Toposes

Modalities, Closure and Reflection



In a context of differential cohesion a reduction modality exhibits an inclusion of its modal types – the reduced objects. Essentially the corresponding inclusion of the anti-modal types is exhibited by an induced modal operator, the relative flat modality rel\flat^{rel}.

Where the plain flat modality \flat sends any object XX to the type X\flat X of its global points, the relative flat modality instead sends it to the type relX\flat^{rel} X of all infinitesimal disks (i.e. the infinitesimal neighbourhoods of all global points) in XX.

See also at differential cohesion and idelic structure.



Given differential cohesion,

& ʃ \array{ \Re &\dashv& \Im &\dashv& \& \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp }

define operations ʃ relʃ^{rel} and rel\flat^{rel} by

ʃ relX(ʃX)XX ʃ^{rel} X \coloneqq (ʃ X) \underset{\Re X}{\coprod} X
relX(X)×XX. \flat^{rel} X \coloneqq (\flat X) \underset{\Im X}{\times} X \,.

Hence ʃ relXʃ^{rel} X makes a homotopy pushout square

X X ʃX ʃ relX \array{ \Re X &\longrightarrow& X \\ \downarrow && \downarrow \\ ʃ X &\longrightarrow& ʃ^{rel} X }

and rel\flat^{rel} makes a homotopy pullback square

relX X X X. \array{ \flat^{rel} X &\longrightarrow& X \\ \downarrow && \downarrow \\ \flat X &\longrightarrow& \Im X } \,.

We call ʃ relʃ^{rel} the relative shape modality and rel\flat^{rel} the relative flat modality.



The relative shape and flat modalities of def.

  1. form an adjoint pair (ʃ rel rel)(ʃ^{rel} \dashv \flat^{rel});

  2. whose (co-)modal types are precisely the properly infinitesimal types, hence those for which \flat \to \Im is an equivalence;

  3. ʃ relʃ^{rel} preserves the terminal object.

It follows that when rel\flat^{rel} has a further right adjoint rel\sharp^{rel} with equivalent modal types containing the codiscrete types, then this defines a level

rel rel * \array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv& \sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast }

hence an intermediate subtopos GrpdH infinitesimalH th\infty Grpd \hookrightarrow \mathbf{H}_{infinitesimal}\hookrightarrow \mathbf{H}_{th} which is infinitesimally cohesive.

This happens notably for the model of formal smooth ∞-groupoids and all its variants such as formal complex analytic ∞-groupoids etc. But in this case ( rel rel)(\flat^{rel} \dashv \sharp^{rel}) does not provide Aufhebung for ()(\flat \dashv \sharp).



The counit of the relative flat modality is a formally étale morphism.

Last revised on May 27, 2015 at 14:13:22. See the history of this page for a list of all contributions to it.