A shadow framework for equivariant Hochschild homologies

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 V, as well as for small V-categories and monoids in V equipped with an automorphism. 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 demonstrate that Hochschild homology of Green functors and Cn-twisted topological Hochschild homology fit into this framework and therefore extend to bicategorical shadows, whence these theories are Morita invariant. Moreover, the associated linearization maps in each case are lax shadow functors. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic K-theory to twisted topological Hochschild homology.