Cyclic homology for bornological coarse spaces

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 $\mathcal{X}H{H}_k^G$ and $\mathcal{X}H{C}_k^G$ 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 $\mathcal{X}{K}_k^G$ 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.