flat modality



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 is also the right adjoint in an adjoint modality with the shape modality \int.


Relation to discrete and codiscrete objects


tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale contractible ʃ discrete discrete differential * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


See the references at local topos.

Revised on November 26, 2014 17:29:36 by Thomas Holder (