nLab infinitesimal shape modality

Idea

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.

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 $X$, i.e. those $(Y\to X) \in \mathbf{H}_{/X}$ for which the naturality square of the $\Im$-unit

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

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

Relation to de Rham spaces

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

Relation to jet bundles

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

Relation to crystalline cohomology

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

Related concepts

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

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

References

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

• Urs Schreiber, Def. 3.5.4 in: differential cohomology in a cohesive topos [arXiv:1310.7930]

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

• Carlos Simpson, Constantin Teleman, Section 3 of: deRham theorem for $\infty$-stacks (1997) [pdf, pdf]

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

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations [arXiv:1701.06238]

reviewed in

• Fosco Loregian, pp. 30 in: Cohesion in Rome, talk notes (Dec. 2019) [pdf, pdf]

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

• Felix Wellen, Cartan Geometry in Modal Homotopy Type Theory (arXiv:1806.05966)

It is called the crystalline modality in

• David Jaz Myers, Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory (arXiv:2205.15887)