infinitesimal shape modality
Modalities, Closure and Reflection
In a context of differential cohesion the infinitesimal shape modality or étale modality characterizes coreduced objects. It is itself the right adjoint in an adjoint modality with the reduction modality and the left adjoint in an adjoint modality with the infinitesimal flat modality.
A context of differential cohesion is determined by the existence of an adjoint triple of modalities
where and are idempotent comonads and is an idempotent monad.
Here is the infinitesimal shape modality. The reflective subcategory that it defines is that of coreduced objects.
Relation for formally étale morphisms
The modal types of in the context of some , i.e. those for which the naturality square of the -unit
is a (homotopy) pullback square, are the formally étale morphisms .
Relation to de Rham spaces
For a geometric homotopy type, the result of applying the infinitesimal shape modality yields a type which has the interpretation of the de Rham space of . See there for more.
Relation to jet bundles
For any object in differential cohesion, the base change comonad along the unit has the interpretation of being the jet comonad which sends bundles over to their jet bundles.
Relation to crystalline cohomology
The cohomology of has the interpretation of crystalline cohomology of . See there for more.
graded differential cohesion
Revised on November 30, 2016 15:05:45
by Urs Schreiber