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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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Last revised on March 5, 2015 at 14:32:17. See the history of this page for a list of all contributions to it.