A Shadow Perspective on Equivariant Hochschild Homologies

Author:

Adamyk Katharine1,Gerhardt Teena2,Hess Kathryn3,Klang Inbar4,Kong Hana Jia5

Affiliation:

1. Hamline University , St Paul, MN 55104, USA

2. Department of Mathmematics , Michigan State University, East Lansing, MI 48824, USA

3. SV UPHESS BMI , École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

4. Department of Mathematics , Columbia University, New York, NY 10027, USA

5. School of Mathematics , Institute for Advanced Study, Princeton, NJ 08540, USA

Abstract

AbstractShadows 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.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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