Contents

Examples

# 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

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

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

Here $\&$ is the infinitesimal flat modality.

## Properties

### Relation to crystalline cohomology

For $A$ a geometric homotopy type, $\& A$ is the coefficient for crystalline cohomology with coefficients in $A$. 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

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

tangent cohesion

differential cohesion

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