Cyclic homology for bornological coarse spaces

Abstract

The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) from the category \(G\mathbf {BornCoarse}\) of equivariant bornological coarse spaces to the cocomplete stable \(\infty \)-category \(\mathbf {Ch}_\infty \) of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory \(\mathcal {X}K^G_{}\) and to coarse ordinary homology?\({{\,\mathrm{\mathcal {X}H}\,}}^G\) by constructing a trace-like natural transformation \(\mathcal {X}K_{}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G\) that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) with the associated generalized assembly map.