# nLab reduction modality

### Context

#### Cohesion

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $\left(\infty ,1\right)$-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 reduction modality characterizes reduced objects. It forms itself the left 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

$\mathrm{Red}⊣{ʃ}_{\mathrm{inf}}⊣{♭}_{\mathrm{inf}}\phantom{\rule{thinmathspace}{0ex}},$Red \dashv &#643;_{inf} \dashv \flat_{inf} \,,

where $\mathrm{Red}$ and ${♭}_{\mathrm{inf}}$ are idempotent comonads and ${ʃ}_{\mathrm{inf}}$ is an idempotent monad.

Here $\mathrm{Red}$ is the reduction modality. The reflective subcategory that it defines is that of reduced objects.

cohesion

• (shape modality $⊣$ flat modality $⊣$ sharp modality)

$\left(ʃ⊣♭⊣♯\right)$

differential cohesion

Revised on November 4, 2013 22:01:37 by Urs Schreiber (82.169.114.243)