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

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

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