# Cyclic homology for bornological coarse spaces

## TL;DR

This paper introduces Hochschild and cyclic homologies for bornological coarse spaces, establishing their relationships with coarse algebraic K-theory and coarse homology, and analyzing associated assembly maps.

## Contribution

It defines equivariant Hochschild and cyclic homologies for bornological coarse spaces and connects them to coarse K-theory and homology via trace-like transformations.

## Key findings

- Hochschild and cyclic homologies are functorial for bornological coarse spaces.
- A natural transformation relates coarse K-theory to coarse homology through Hochschild and cyclic homologies.
- Comparison of forget-control and assembly maps provides insights into coarse homological invariants.

## Abstract

We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse spaces $G\mathbf{BornCoarse}$ to the cocomplete stable $\infty$-category of chain complexes $\mathbf{Ch}_\infty$. We relate these equivariant coarse homology theories to coarse algebraic $K$-theory $\mathcal{X} K^G_{k}$ and to coarse ordinary homology $\mathcal{X} H^G$ by constructing a trace-like natural transformation $\mathcal{X} K_{k}^G\to \mathcal{X} H^G$ that factors through coarse Hochschild (or cyclic) homology. We further compare the forget-control map for coarse Hochschild homology with the associated generalized assembly map.