structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In a context of synthetic differential geometry/differential cohesion the reduction modality characterizes reduced objects. It forms itself the left 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
where $\Re$ and $\&$ are idempotent comonads and $\Im$ is an idempotent monad, furthermore $\Re$ preserves finite products.
Here $\Re$ is the reduction modality. The reflective subcategory that it defines is that of reduced objects.
(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
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$