The universal Hochschild shadow: from bicategories to $(\infty,2)$- categories

(joint with Nima Rasekh)
The theory of shadows, first introduced by Ponto, is an axiomatic, bicategorical framework that     generalizes (topological) Hochschild homology and satisfies analogous important properties, such as Morita invariance. I’ll explain how to use Berman's extension of Hochschild homology to bicategories to prove that there is an equivalence between functors out of the Hochschild homology of a bicategory and shadows on that bicategory, from which it follows that Hochschild homology of bicategories actually provides a universal shadow on bicategories and which enables to formulate Morita invariance functorially. I’ll then describe the infinity categorical generalization of this story, parts of which are still conjectural.