A topological ∞-groupoid is an internal ∞-groupoid in Top. As described there, a general interpretation of what that means is to say that this is an ∞-stack on Top.
The sub-(∞,1)-category ∞-stacks on Top (even on Diff) that are homotopy invariant is equivalent to plain ∞Grpd.
A central result about the (∞,1)-topos of ∞-stacks on Top is that the homotopy localization is equivalent to Top itself
A discussion of this is in (the nice but not quite finished)
In fact, this is true even for Lie ∞-groupoids, i.e. ∞-stacks on Diff: the homotopy invariant ones model plain topological spaces.
This provides a useful resolution of topological spaces that often helps to disentangle the two different roles played by a topological space: on the one hand as a model for an ∞-groupoid, in the other as a locale.
Let be the local model structure on simplicial presheaves obtained by left Bousfield localization at the Cech nerves of Cech covers with respect to the standard Grothendieck topology on Diff. This is a model for ∞-stacks on Diff.
Let be furthermore the left Bousfield localization at the set of projection morphisms out of products of the form for all . The -stacks that are local objects with respect to these morphisms are the homotopy invariant -stacks, so this localization models the (∞,1)-topos of homotopy invariant -stacks on .
There is a adjunction
where sends a simplicial set to the simplicial presheaf constant on that simplicial set, and where evaluates a simplicial presheaf on the manifold that is the point.
Theorem (Dugger)
This adjunction is a Quillen equivalence with respect to the standard model structure on simplicial sets on the left and the above model structure on the right.