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 by definition classifies the ∞-geometric morphisms into it in that it is the representing object of .
A special case of this is the is the notion of a classifying (∞,1)-topos for a geometry in the sense of structured spaces:
The geometry is the (∞,1)-category that plays role of the syntactic theory. For 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 in the (Cech) ∞-stack (∞,1)-topos and exhibits it as the classifying topos for such models (geometries):
This is Structured Spaces prop 1.4.2.