nLab
codiscrete object

Definition

For $\Gamma :ℰ\to ℬ$ a functor we say that it has codiscrete objects if it has a full and faithful right adjoint $\mathrm{coDisc}:ℬ\hookrightarrow ℰ$.

An object in the essential image of $\mathrm{coDisc}$ is called a codiscrete object.

This is for instance the case for the global section geometric morphism of a local topos $(\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc})ℰ\to ℬ$.

If one thinks of $ℰ$ as a category of spaces, then the codiscrete objects are called codiscrete spaces.

The dual notion is that of discrete objects.

Properties

$\Gamma$ is a faithful functor on morphisms whose codomain is concrete.

Properties

Proposition

If $ℰ$ has a terminal object that is preserved by $\Gamma$, then $ℰ$ has concrete objects.

This is (Shulman, theorem 1).

Proposition

If $ℰ$ has codiscrete objects and has pullbacks that are preserved by $\Gamma$ and , then $\Gamma$ is a Grothendieck fibration.

This is (Shulman, theorem 2).

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

References

• Mike Shulman, Discreteness, Concreteness, Fibrations, and Scones (blog post)