K-homology of C ^*-categories and symmetric spectra representing K-homology

Abstract

We define K-homology groups K _* (𝒞) for small C ^*-categories 𝒞 in terms of Hilbert modules over the C ^*-category 𝒞. We also define a functor A ^f from the category of small C ^*-categories into the category of C ^*-algebras and show that there is a natural isomorphism $K_*(\cal C)\cong K_*(A^f_{\cal C})$. In addition, we give an easy construction of a functor ${\mathbb K}$ from the category of C ^*-algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor ${\mathbb K}$ comes with a natural isomorphism $K_*(A)\cong \pi_*({\mathbb K}(A))$ for C ^*-algebras A. It then follows that the composition ${\mathbb K}$○A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.