structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
typical contexts
In a context of synthetic differential geometry/differential cohesion a coreduced object is one all whose infinitesimal paths are constant. Compare the discrete objects, in which all paths are constant, meaning all discrete objects are also coreduced.
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.
A coreduced object or coreduced type is one in the full subcategory defined by the infinitesimal shape modality $\Im$ or equivalently the infinitesimal flat modality $\&$.
Note that an object $X$ being coreduced is the same as it being formally etale.
(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
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$