nLab
flat modality

Contents

Definition

On a local topos/local (∞,1)-topos H\mathbf{H}, hence with extra fully faithful right adjoint coDisccoDisc to the global section geometric morphism (DiscΓ)(Disc \dashv \Gamma), is canonically induced the idempotent comonad DiscΓ\flat \coloneqq Disc\circ \Gamma. This modality sends for instance pointed connected objects BG\mathbf{B}G to coefficients BG\flat \mathbf{B}G 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 coDiscΓ\sharp \coloneqq coDisc \circ \Gamma. If H\mathbf{H} is in addition a cohesive (∞,1)-topos then it also the right adjoint in an adjoint modality with the shape modality \int.

Properties

Relation to discrete and codiscrete objects

cohesion

tangent cohesion

differential cohesion

References

See the references at local topos.

Revised on November 4, 2013 21:58:29 by Urs Schreiber (82.169.114.243)