structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
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.
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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR}\dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv R)$