Hochschild homology was originally introduced as an invariant of associative algebras that generalizes the commutator. In subsequent decades, it has been extended both in categorical directions, such as to dg-categories, and in topological ones, such as to ring spectra. These generalizations have also enabled many fruitful interactions with other theories, such as algebraic K-theory or fixed point methods. The existence of Hochschild homology in such diverse contexts motivated Ponto to introduce a notion that can encompass the various perspectives: a shadow of bicategories. In this talk, I will show that shadows themselves admit a characterization in terms of a bicategorical version of Hochschild homology. I will then discuss various powerful applications of this perspective, such as \u221e-categorical versions of shadows, Morita invariance properties, and Hochschild homology of enriched bimodules. Joint work with Kathryn Hess.
Recording:<\/p>\n\n\n\n
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