Higher generalized morphisms and Morita equivalence of geometric $\infty$-groupoids

I will carefully review the various equivalent definitions of principal $G$-bundles (free and proper $G$-action, fiber bundle with free and transitive $G$-action on fibers, local transition functions on cover satisfying cocycle condition, classifying map to $BG$). Then I show how these notions generalize to Lie $\infty$-groupoids. The main result is that we can still move between principal groupoid bibundles, anafunctors, and classifying maps in the $\infty$-categorical setting. This implies that we have a consistent notion of Morita equivalence that provides a localization of Lie $\infty$-groupoids at weak equivalences. Our approach also works for other "geometric" categories such as Banach manifolds, topological spaces, diffeological spaces, affine schemes, etc. that are equipped with a Grothendieck topology satisfying a number of axioms. This is joint work with Chenchang Zhu and Kalin Krishna.