nLab
coreduced object

Context

Cohesion

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Discrete and concrete objects

Modalities, Closure and Reflection

Contents

Idea

In a context of synthetic differential geometry/differential cohesion a coreduced object is one all whose infinitesimal paths are constant. Compare the discrete objects, in which all paths are constant, meaning all discrete objects are also coreduced.

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.

A coreduced object or coreduced type is one in the full subcategory defined by the infinitesimal shape modality \Im or equivalently the infinitesimal flat modality &\&.

Note that an object XX being coreduced is the same as it being formally etale.

Examples

cohesion

tangent cohesion

differential cohesion

graded differential 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& Rh & \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 }

Revised on August 29, 2017 12:19:24 by Max S. New (129.10.110.48)