nLab coreduced object

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 $X$ being coreduced is the same as it being formally etale.

Examples

• A coreduced scheme is also called a de Rham space. The cohomology over such is crystalline cohomology, the quasicoherent sheaves over these define D-geometry.
• In a differential geometric setting, formally etale maps are the local diffeomorphisms, and the coreduced objects are those for which the terminal map $X \to 1$ is a local diffeomorphism.

Related concepts

cohesion

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

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

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

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

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

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

singular cohesion

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

• formally smooth morphism, formally etale morphism, formally unramified morphism