(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of a classifying (∞,1)-topos is the vertical categorification of the notion of classifying topos to the context of (∞,1)-category theory.
Any (∞,1)-topos $K$ by definition classifies the ∞-geometric morphisms into it in that it is the representing object of $geom(-,K)$.
Write $\infty Grpd_{fin}$ for the (∞,1)-category of finite homotopy types and write $\infty Grpd_{fin}^{\ast/}$ for its pointed objects.
For $\mathbf{H}$ any base (∞,1)-topos, the (∞,1)-presheaf (∞,1)-topos
is the classifying $\infty$-topos for objects, and
is the classifying $\infty$-topos for pointed objects.
For $K \in \infty Grpd_{fin}^{\ast/}$, write $R(K) \in (\infty Grpd_{fin}^{\ast/})^{op} \hookrightarrow \mathbf{H}[X_\ast]$ for its formal dual under (∞,1)-Yoneda embedding. The generic pointed object in $\mathbf{H}[X_\ast]$ is that represented by the 0-sphere:
A special case of this is the notion of a classifying (∞,1)-topos for a geometry in the sense of structured spaces:
The geometry $\mathcal{G}$ is the (∞,1)-category that plays role of the syntactic theory. For $\mathcal{X}$ an (∞,1)-topos, a model of this theory is a limits and covering-preserving (∞,1)-functor
The Yoneda embedding followed by ∞-stackification
constitutes a model of $\mathcal{G}$ in the (Cech) ∞-stack (∞,1)-topos $Sh_{(\infty,1)}(\mathcal{G})$ and exhibits it as the classifying topos for such models (geometries):
This is Structured Spaces prop 1.4.2.