nLab
co-concrete object

Definition

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

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

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

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

The dual notion is that of codiscrete objects.

References

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