# Contents

## Definition

On a local topos/local (∞,1)-topos $\mathbf{H}$, hence equipped with a fully faithful extra right adjoint $coDisc$ to the global section geometric morphism $(Disc \dashv \Gamma)$, is induced an idempotent monad $\sharp \coloneqq coDisc \circ \Gamma$, a modality which we call the sharp modality. This is itself the right adjoint in an adjoint modality with the flat modality $\flat \coloneqq Disc \circ \Gamma$.

## Properties

### Relation to discrete and codiscrete objects

cohesion

tangent cohesion

differential cohesion

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