nLab co-concrete object

Definition

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

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

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

In this situation, we say that a co-concrete object $X \in \mathcal{E}$ is one for which the $(Disc\dashv \Gamma)$-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)