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Idea

In a context of synthetic differential geometry/differential cohesion the reduction modality characterizes reduced objects. It forms itself the left adjoint in an adjoint modality with the infinitesimal shape modality.

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 . , furthermore$\Re$ preserves finite products.

Here $\Re$ is the reduction modality. The reflective subcategory that it defines is that of reduced objects.

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