The purpose of this paper is to give a detailed study of the basic theory of C*‐categories. The study includes some examples of C*‐categories that occur naturally in geometric applications, such as groupoid C*‐categories, and C*‐categories associated to structures in coarse geometry. We conclude the paper with a brief survey of Hilbert modules over C*‐categories. 2000 Mathematical Subject Classification: 18D99, 46L05, 46L08.
In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of "coarse CW -complexes". This uniqueness result is used to prove a version of the coarse Baum-Connes conjecture for such spaces.
In this article, we give a characterisation of the Baum-Connes assembly map with coefficients. The technical tools needed are the K-theory of C Ã -categories, and equivariant KK-theory in the world of groupoids. r
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive-a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an associated coarse assembly map from the topologically exicisive functor to the coarsely excisive functor.We conjecture that this coarse assembly map is an isomorphism for uniformly contractible spaces with bounded geometry, and show that the coarse isomorphism conjecture, along with some mild technical conditions, implies that a correspoding equivariant assembly map is injective. Particular instances of this equivariant assembly map are the maps in the Farrell-Jones conjecture, and in the Baum-Connes conjecture.
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