Shadows are Bicategorical Traces (2109.02144v3)

Abstract: Hochschild homology has proved to be an important invariant in algebra and homotopy theory, in particular due to its relevance in algebraic $K$-theory and fixed point theory, leading to the development of numerous variants of the original construction. Ponto introduced a bicategorical axiomatization of Hochschild homology-type invariants, called a shadow, which captures the essential common properties of all known variants of Hochschild homology, such as Morita invariance. In this paper we clarify the relationship between shadows and Hochschild homology. After extending the notion of Hochschild homology to bicategories in a natural manner, we prove the existence of a universal shadow on any bicategory $\mathscr{B}$, taking values in the Hochschild homology of $\mathscr{B}$, through which all other shadows on $\mathscr{B}$ factor. Shadows are thus co-represented by a bicategorical version of Hochschild homology. Using the universal shadow on the free adjunction bicategory, we can then establish a universal Morita invariance theorem, of which all known cases are immediate corollaries. Building on this understanding of shadows on bicategories, we propose an $\infty$-categorical generalization of shadows as functors out of Hochschild homology of an $(\infty,2)$-category in the sense of Berman. As a first step towards constructing relevant examples of $\infty$-categorical shadows, we define the Hochschild homology of enriched $\infty$-categorical bimodules and prove that they assemble into a shadow. As part of this work we compute the Hochschild homology of several important $2$-categories (such as the free adjunction), which can be of independent interest.