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

ʃ ʃ inf inf * \array{ && && \Re \\ && && \bot \\ && ʃ & \subset & ʃ_{inf} \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \flat_{inf} \\ \bot & & \bot && \\ \ast & \subset& \sharp }
Revised on November 5, 2013 02:39:42 by Urs Schreiber (145.116.129.122)