nLab formally smooth object

Contents

Context

Cohesion

Discrete and concrete objects

Modalities, Closure and Reflection

Contents

Definition

A context of differential cohesion is determined by the existence of an adjoint triple of modalities

&, \Re \dashv \Im \dashv \& \,,

where \Re and &\& are idempotent comonads and \Im is an idempotent monad. Here \Im is called the infinitesimal shape modality.

An object/type XX is called formally smooth if the unit

XX X \to \Im X

is a 1-epimorphism. This is equivalent to the essentially unique morphism X*X \to * to the terminal object being a formally smooth morphism.

Examples

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Last revised on August 24, 2018 at 12:13:03. See the history of this page for a list of all contributions to it.