A shadow perspective on equivariant Hochschild homologies

TL;DR

This paper develops a generalized framework for Hochschild homology using shadows in bicategories, establishing Morita invariance and connecting algebraic and topological theories through new linearization maps and a twisted Dennis trace map.

Contribution

It introduces Hochschild-type invariants for monoids and categories in symmetric monoidal model categories, extending to shadows and proving Morita invariance, including twisted variants and new linearization maps.

Findings

Hochschild invariants are Morita invariant in this framework.

The linearization map for topological Hochschild homology is a lax shadow functor.

A twisted Dennis trace map from equivariant K-theory to twisted topological Hochschild homology is constructed.

Abstract

Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.