nLab
flat modality

Definition

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

Properties

Relation to discrete and codiscrete objects

• The codiscrete objects in a local topos are the modal types for the sharp modality.

• The discrete objects in a local topos are the modal types for the flat modality.

Related concepts

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(ʃ \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $ʃ_{dR} \dashv \flat_{dR}$

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

References

See the references at local topos.