Local admissibility structure | Higher Topos Theory notes

Definition 7.18.

Let $$E$$ be a topos. An admissibility structure $E^{\ad}$ is called local if the following conditions are satisfied:

The class $E^{\ad}$ is a local class of maps.
For every $X \in E$ , the slice $E^{\ad}_{/X} := (E^{\ad})_{/X}$ is presentable, and the inclusion $E^{\ad}_{/X} \to E_{/X}$ preserves colimits. In particular, it admits a right adjoint $\rho_X\colon E_{/X} \to E^{\ad}_{/X}$ .

Given a local admissibility structure on $$E$$ , we say that an object $X \in E$ is corporeal if the functor $\rho_X\colon E_{/X} \to E^{\ad}_{/X}$ preserves colimits. We denote by

E^{\corp} \subseteq E^{\ad}

the full subcategory spanned by the corporeal objects. We say that $E^{\ad}$ is geometric if, moreover, $$E$$ is generated under colimits by corporeal objects.