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

Redʃ inf inf, Red \dashv ʃ_{inf} \dashv \flat_{inf} \,,

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

Here inf\flat_{inf} is the infinitesimal flat modality.

Properties

Relation to crystalline cohomology

For AA a geometric homotopy type, infA\flat_{inf} A is the coefficient for crystalline cohomology with coefficients in AA. See there for more.

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 November 5, 2013 02:39:42 by Urs Schreiber (145.116.129.122)