# nLab infinitesimal shape modality

### Context

#### Cohesion

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

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

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

# Contents

## Idea

In a context of differential cohesion the infinitesimal shape modality 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

$Red \dashv ʃ_{inf} \dashv \flat_{inf} \,,$

where $Red$ and $\flat_{inf}$ are idempotent comonads and $ʃ_{inf}$ is an idempotent monad.

Here $ʃ_{inf}$ is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.

## Properties

### Relation to de Rham spaces

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

### Relation to jet bundles

For $E$ a dependent type on $X$, its dependent product along the unit of the infinitesimal shape modality has the interpretation of the jet type $j(E)$. See there for more.

### Relation to crystalline cohomology

The cohomology of $ʃ_{inf} X$ has the interpretation of crystalline cohomology of $X$. See there for more.

cohesion

tangent cohesion

differential cohesion

$\array{ id & \dashv & id \\ \vee && \vee \\ \Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }$
Revised on April 24, 2014 23:02:04 by David Corfield (46.208.3.134)