nLab
reduction modality

Idea

In a context of synthetic differential geometry/differential cohesion the reducton modality characterizes reduced objects.

Definition

A context of differential cohesion is determined by the existence of an adjoint triple of modalities

$$\mathrm{Red}⊣{ʃ}_{\inf }⊣{♭}_{\inf },$$Red \dashv ʃ_{inf} \dashv \flat_{inf} \,,

where $\mathrm{Red}$ and ${♭}_{\inf }$ are idempotent comonads and ${ʃ}_{\inf }$ is an idempotent monad.

Here $\mathrm{Red}$ is the reduction modality. The reflective subcategory that it defines is that of reduced objects.

Related concepts

cohesion

• (shape modality $⊣$ flat modality $⊣$ sharp modality)

  $(ʃ⊣♭⊣♯)$

  • discrete object, codiscrete object, concrete object

  • structures in cohesion

differential cohesion

• (reduction modality $⊣$ infinitesimal shape modality $⊣$ infinitesimal flat modality)

  $(\mathrm{Red}⊣{ʃ}_{\inf }⊣{♭}_{\inf })$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion