Cyclic homology for bornological coarse spaces

Abstract

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