nLab relative flat modality

Idea

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 $\flat^{rel}$.

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

See also at differential cohesion and idelic structure.

Definition

Definition

Given differential cohesion,

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

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

$$ʃ^{rel} X \coloneqq (ʃ X) \underset{\Re X}{\coprod} X$$

$$\flat^{rel} X \coloneqq (\flat X) \underset{\Im X}{\times} X \,.$$

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

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

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

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

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

Properties

Proposition

The relative shape and flat modalities of def.

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

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

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

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

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

hence an intermediate subtopos $\infty Grpd \hookrightarrow \mathbf{H}_{infinitesimal}\hookrightarrow \mathbf{H}_{th}$ 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 $(\flat^{rel} \dashv \sharp^{rel})$ does not provide Aufhebung for $(\flat \dashv \sharp)$.

(…)

Proposition

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