structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In a context of synthetic differential geometry/differential cohesion the infinitesimal flat modality is the right adjoint in an adjoint modality with the infinitesimal shape modality.
A context of differential cohesion is determined by the existence of an adjoint triple of modalities forming two pairs of adjoint modalities
where $Red$ and $\flat_{inf}$ are idempotent comonads and $ʃ_{inf}$ is an idempotent monad.
Here $\flat_{inf}$ is the infinitesimal flat modality.
For $A$ a geometric homotopy type, $\flat_{inf} A$ is the coefficient for crystalline cohomology with coefficients in $A$. 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)
$(\Re \dashv ʃ_{inf} \dashv \flat_{inf})$