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 Gmathbf {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_{}^Grightarrow {{,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.