nLab infinitesimal shape modality




Modalities, Closure and Reflection



In a context of differential cohesion the infinitesimal shape modality, crystalline 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.


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.


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.


infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


The infinitesimal shape modality as part of an extension of cohesive \infty -toposes to differential cohesive \infty -toposes was introduced in

mimicking an analogous construction (cf. de Rham space) for schemes in

and further developed (in relation to jet comonads and partial differential equations) in

reviewed in

Its implementation in HoTT and application to Cartan geometry is considered in

It is called the crystalline modality in

Last revised on December 5, 2022 at 16:41:43. See the history of this page for a list of all contributions to it.