nLab
codiscrete object

Contents

Definition

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

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

This is for instance the case for the global section geometric morphism of a local topos (DiscΓcoDisc) (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.

Properties

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

Properties

Proposition

If \mathcal{E} has a terminal object that is preserved by Γ\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 Γ\Gamma and , then Γ\Gamma is a Grothendieck fibration.

This is (Shulman, theorem 2).

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic R rheonomic reduced infinitesimal infinitesimal & étale contractible ʃ discrete discrete differential * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \R & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

References

Revised on January 5, 2013 21:56:47 by Urs Schreiber (89.204.138.93)