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

graded differential cohesion

id id fermionic e bosonic bosonic R rheonomic reduced infinitesimal infinitesimal & étale contractible ʃ discrete discrete differential * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \e &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \R & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Revised on March 3, 2015 10:25:33 by Urs Schreiber (80.92.246.195)