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 or étale modality \Im 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

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

where \Re and &\& are idempotent comonads and \Im is an idempotent monad.

Here \Im is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.

Properties

Relation for formally étale morphisms

The modal types of \Im in the context of some XX, i.e. those (YX)H /X(Y\to X) \in \mathbf{H}_{/X} for which the naturality square of the \Im -unit

Y Y X X \array{ Y &\longrightarrow& \Im Y \\ \downarrow && \downarrow \\ X &\longrightarrow& \Im X }

is a (homotopy) pullback square, are the formally étale morphisms YXY \to X.

Relation to de Rham spaces

For XX a geometric homotopy type, the result of applying the infinitesimal shape modality yields a type X \Im X which has the interpretation of the de Rham space of XX. See there for more.

Relation to jet bundles

For any object XX in differential cohesion, the base change comonad Jeti *i *Jet \coloneqq i^\ast i_\ast along the unit i:XXi \colon X \to \Im X has the interpretation of being the jet comonad which sends bundles over XX to their jet bundles.

Relation to crystalline cohomology

The cohomology of X\Im X has the interpretation of crystalline cohomology of XX. See there for more.

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic R 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& \R & \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 May 15, 2015 10:59:26 by Urs Schreiber (195.113.30.252)