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 $\Re$ and $\&$ are idempotent comonads and $\Im$ is an idempotent monad.
Here $\&$ is the infinitesimal flat modality.
For $A$ a geometric homotopy type, $\& A$ 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)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\e \dashv \rightsquigarrow \dashv R)$