co-concrete object in nLab

Contents

Contents

Definition

For Γ:ℰ→ℬ\Gamma : \mathcal{E} \to \mathcal{B} a functor we say that it has discrete objects if it has a full and faithful left adjoint Disc:ℬ↪ℰDisc : \mathcal{B} \hookrightarrow \mathcal{E}.

An object in the essential image of DiscDisc is called a discrete object.

This is for instance the case for the global section geometric morphism of a connected topos (Disc⊣Γ):ℰ→ℬ (Disc \dashv \Gamma ) : \mathcal{E} \to \mathcal{B}.

In this situation, we say that a co-concrete object X∈ℰX \in \mathcal{E} is one for which the (Disc⊣Γ)(Disc\dashv \Gamma)-counit of an adjunction is an epimorphism.

The dual concept is the of a concrete object.

References

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