nLab
concrete 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}.

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}.

In this situation, we say that a concrete object XX \in \mathcal{E} is one for which the (ΓcoDisc)(\Gamma \dashv coDisc)-unit of an adjunction is a monomorphism.

If \mathcal{E} is a sheaf topos, this is called a concrete sheaf.

If \mathcal{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

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

cohesion

tangent cohesion

differential cohesion

id id ʃ inf inf ʃ * \array{ id & \dashv & id \\ \vee && \vee \\ \Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }

References

Revised on February 20, 2013 10:37:24 by Urs Schreiber (80.81.16.253)