nLab
infinitesimal flat 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 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 AA a geometric homotopy type, &A\& A is the coefficient for crystalline cohomology with coefficients in AA. See there for more.

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale contractible ʃ discrete discrete differential * \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{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }
Revised on March 5, 2015 14:32:17 by Urs Schreiber (195.113.30.252)