nLab
concrete object

Definition

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

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

In this situation, we say that a concrete object $X\in ℰ$ is one for which the $(\Gamma ⊣\mathrm{coDisc})$-unit of an adjunction is a monomorphism.

If $ℰ$ is a sheaf topos, this is called a concrete sheaf.

If $ℰ$ is a cohesive (∞,1)-topos then this is called a concrete (∞,1)-sheaf or the like.

The dual notion is that of a co-concrete object.

Properties

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

Related concepts

• concretification

cohesion

• (shape modality $⊣$ flat modality $⊣$ sharp modality)

  $(ʃ⊣♭⊣♯)$

  • discrete object, codiscrete object, concrete object

  • structures in cohesion

differential cohesion

• (reduction modality $⊣$ infinitesimal shape modality $⊣$ infinitesimal flat modality)

  $(\mathrm{Red}⊣{ʃ}_{\inf }⊣{♭}_{\inf })$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

References

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