nLab
reduction modality

Context

Cohesion

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Modalities, Closure and Reflection

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

ʃ inf inf, \Re \dashv ʃ_{inf} \dashv \flat_{inf} \,,

where \Re and inf\flat_{inf} are idempotent comonads and ʃ infʃ_{inf} is an idempotent monad.

Here \Re is the reduction modality. The reflective subcategory that it defines is that of reduced objects.

cohesion

tangent cohesion

differential cohesion

id id ʃ inf inf ʃ * \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 August 18, 2014 12:26:57 by Urs Schreiber (89.15.239.24)