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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
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.
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
is a (homotopy) pullback square, are the formally étale morphisms $Y \to X$.
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.
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.
The cohomology of $\Im X$ has the interpretation of crystalline cohomology of $X$. See there for more.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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.