nLab reduction 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 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

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

where $\Re$ and $\&$ are idempotent comonads and $\Im$ is an idempotent monad, furthermore $\Re$ preserves finite products.

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

cohesion

tangent cohesion

differential cohesion

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