# nLab infinitesimal flat 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 synthetic differential geometry/differential cohesion the infinitesimal flat modality is the right adjoint in an adjoint modality with the infinitesimal shape modality.

## Definition

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

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

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

Here $\flat_{inf}$ is the infinitesimal flat modality.

## Properties

### Relation to crystalline cohomology

For $A$ a geometric homotopy type, $\flat_{inf} A$ is the coefficient for crystalline cohomology with coefficients in $A$. See there for more.

cohesion

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

$(ʃ \dashv \flat \dashv \sharp )$

tangent cohesion

differential cohesion

Revised on November 5, 2013 02:39:42 by Urs Schreiber (145.116.129.122)