nLab codiscrete object

Definition

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

An object in the essential image of $coDisc$ is called a codiscrete object.

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

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

The dual notion is that of discrete objects.

Examples

• codiscrete topology

• codiscrete groupoid

Properties

Proposition

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

Proposition

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

This is Shulman, theorem 1.

Proposition

If $\mathcal{E}$ 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

• chaos

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

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

References

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