Contents

Examples

# Contents

## Idea

In a context of differential cohesion the infinitesimal shape modality, crystalline modality or étale modality $\Im$ characterizes coreduced objects. It is itself the right adjoint in an adjoint modality with the reduction modality and the left adjoint in an adjoint modality with the infinitesimal flat modality.

## Definition

A context of differential cohesion is determined by the existence of an adjoint triple of modalities

$\Re \dashv \Im \dashv \& \,,$

where $\Re$ and $\&$ are idempotent comonads and $\Im$ is an idempotent monad.

Here $\Im$ is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.

## Properties

### Relation for formally étale morphisms

The modal types of $\Im$ in the context of some $X$, i.e. those $(Y\to X) \in \mathbf{H}_{/X}$ for which the naturality square of the $\Im$-unit

$\array{ Y &\longrightarrow& \Im Y \\ \downarrow && \downarrow \\ X &\longrightarrow& \Im X }$

is a (homotopy) pullback square, are the formally étale morphisms $Y \to X$.

### Relation to de Rham spaces

For $X$ a geometric homotopy type, the result of applying the infinitesimal shape modality yields a type $\Im X$ which has the interpretation of the de Rham space of $X$. See there for more.

### Relation to jet bundles

For any object $X$ in differential cohesion, the base change comonad $Jet \coloneqq i^\ast i_\ast$ along the unit $i \colon X \to \Im X$ has the interpretation of being the jet comonad which sends bundles over $X$ to their jet bundles.

### Relation to crystalline cohomology

The cohomology of $\Im X$ has the interpretation of crystalline cohomology of $X$. See there for more.

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

The infinitesimal shape modality as part of an extension of cohesive $\infty$-toposes to differential cohesive $\infty$-toposes was introduced in

mimicking an analogous construction (cf. de Rham space) for schemes in

and further developed (in relation to jet comonads and partial differential equations) in

reviewed in

Its implementation in HoTT and application to Cartan geometry is considered in

It is called the crystalline modality in