On a local topos/local (∞,1)-topos , hence with extra fully faithful right adjoint to the global section geometric morphism , is canonically induced the idempotent comonad . This modality sends for instance pointed connected objects to coefficients for flat principal ∞-connections, and may therefore be referred to as the flat modality. It is itself the left adjoint in an adjoint modality with the sharp modality . If is in addition a cohesive (∞,1)-topos then it also the right adjoint in an adjoint modality with the shape modality .
See the references at local topos.