nLab
infinitesimal shape 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 differential cohesion the infinitesimal shape modality characterizes coreduced objects. It is itself the right adjoint in an adjoint modality with the reduction modality and the left adjoint in an adjoint modality with the infinitesimal flat modality.

Definition

A context of differential cohesion is determined by the existence of an adjoint triple of modalities

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

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

Here ʃ infʃ_{inf} is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.

Properties

Relation to de Rham spaces

For XX a geometric homotopy type, the result of applying the infinitesimal shape modality yields a type ʃ infXʃ_{inf} X which has the interpretation of the de Rham space of XX. See there for more.

Relation to jet bundles

For EE a dependent type on XX, its dependent product along the unti of the infinitesimal shape modality has the interpretation of the jet type j(E)j(E). See there for more.

Relation to crystalline cohomology

The cohomology of ʃ infXʃ_{inf} X has the interpretation of crystalline cohomology of XX. See there for more.

cohesion

differential cohesion

Revised on November 5, 2013 02:21:31 by Urs Schreiber (145.116.129.122)