structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In a context of differential cohesion the infinitesimal shape modality or étale 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.
A context of differential cohesion is determined by the existence of an adjoint triple of modalities
where $\Re$ and $\Im$ are idempotent comonads and $\&$ is an idempotent monad.
Here $\&$ is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.
The modal types of $\&$ in the context of some $X$, i.e. those $(Y\to X) \in \mathbf{H}_{/Y}$ for which the naturality square of the $\&$-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 $\& X$ which has the interpretation of the de Rham space of $X$. See there for more.
For $E$ a dependent type on $X$, its dependent product along the unit of the infinitesimal shape modality has the interpretation of the jet type $j(E)$. See there for more.
The cohomology of $\& X$ has the interpretation of crystalline cohomology of $X$. See there for more.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR}\dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \& \dashv \Im)$