coreduced object in nLab

Contents

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

ℜ⊣ℑ⊣&, \Re \dashv \Im \dashv \& \,,

where ℜ\Re and &\& are idempotent comonads and ℑ\Im is an idempotent monad.

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

Note that an object XX 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→1X \to 1 is a local diffeomorphism.

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