Modalities, Closure and Reflection
On a local topos/local (∞,1)-topos , hence equipped with a fully faithful extra right adjoint to the global section geometric morphism , is induced an idempotent monad , a modality which we call the sharp modality. This is itself the right adjoint in an adjoint modality with the flat modality .
Relation to discrete and codiscrete objects
graded differential cohesion
See the references at local topos.