Spahn locally representable structured (infinity,1)-topos (Rev #1)

Structured $(\infty,1)$-topos $(X,O_{G,X})$ where

(1) $X$ is an $(\infty,1)$-topos

(2) $G=(\underline G, Ad_G)$ is a geometry, i.e. $\underline G$ is an essentially small $(\infty,1)$-category with finite limits, which is idempotent complete, $Ad_G$ is an admissible structure in $\underline G$.

(3) $O_{G,X}:G\to X$ is a $G$-structure (aka “structure sheaf”) on $X$, i.e. a functor which is

(3a) left exact

(3b) induces a jointly epimorphic family in the image of $O_{G,X}$ (i.e. every covering sieve consisting of admissible morphisms on an object of $X$ induces an effective epimorphism out of the product of the images…)