(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
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 by definition classifies the ∞-geometric morphisms into it in that it is the representing object of .
For objects and pointed objects
Write for the (∞,1)-category of finite homotopy types and write for its pointed objects.
For any base (∞,1)-topos, the (∞,1)-presheaf (∞,1)-topos
is the classifying -topos for objects, and
is the classifying -topos for pointed objects.
For , write for its formal dual under (∞,1)-Yoneda embedding. The generic pointed object in is that represented by the 0-sphere:
For local structures
A special case of this 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.