structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In a context of differential cohesion the infinitesimal shape 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 $Red$ and $\flat_{inf}$ are idempotent comonads adn $ʃ_{inf}$ is an idempotent monad.
Here $ʃ_{inf}$ is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.
For $X$ a geometric homotopy type, the result of applying the infinitesimal shape modality yields a type $ʃ_{inf} X$ 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 unti of the infinitesimal shape modality has the interpretation of the jet type $j(E)$. See there for more.
The cohomology of $ʃ_{inf} X$ has the interpretation of crystalline cohomology of $X$. See there for more.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(Red \dashv ʃ_{inf} \dashv \flat_{inf})$