typical contexts
For $\Gamma : \mathcal{E} \to \mathcal{B}$ a functor we say that it has codiscrete objects if it has a full and faithful right adjoint $coDisc : \mathcal{B} \hookrightarrow \mathcal{E}$.
This is for instance the case for the global section geometric morphism of a local topos $(Disc \dashv \Gamma \dashv coDisc) \mathcal{E} \to \mathcal{B}$.
In this situation, we say that a concrete object $X \in \mathcal{E}$ is one for which the $(\Gamma \dashv coDisc)$-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.
$\Gamma$ is a faithful functor on morphisms whose codomain is concrete.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR}\dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \& \dashv \Im)$