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Categorical aspects of bivariant K-theoryRalf Meyer1 IntroductionNon-commutative topology deals with topological properties of C -algebras. Alreadyin the 1970s, the classification of AF-algebras by K-theoretic data [15] and the workof Brown–Douglas–Fillmore on essentially normal operators [6] showed clearly thattopology provides useful tools to study C -algebras. A breakthrough was Kasparov’sconstruction of a bivariant K-theory for separable C -algebras. Besides its applicationswithin C -algebra theory, it also yields results in classical topology that are hard or evenimpossible to prove without it. A typical example is the Novikov conjecture, whichdeals with the homotopy invariance of certain invariants of smooth manifolds with agiven fundamental group. This conjecture has been verified for many groups usingKasparov theory, starting with [31]. The C -algebraic formulation of the Novikovconjecture is closely related to the Baum–Connes conjecture, which deals with thecomputation of the K-theory K .C red G/ of reduced group C -algebras and has beenone of the centres of attention in non-commutative topology in recent years.The Baum–Connes conjecture in its original formulation [4] only deals with a singleK-theory group; but a better understanding requires a different point of view. Theapproach by Davis and Lück in [14] views it as a natural transformation between twohomology theories for G-CW-complexes. An analogous approach in the C -algebraframework appeared in [38]. These approaches to the Baum–Connes conjecture showthe importance of studying not just single C -algebras, but categories of C -algebrasand their properties. Older ideas like the universal property of Kasparov theory areof the same nature. Studying categories of objects instead of individual objects isbecoming more and more important in algebraic topology and algebraic geometry aswell.Several mathematicians have suggested, therefore, to apply general constructionswith categories (with additional structure) like generators, Witt groups, the centre, andsupport varieties to the C -algebra context. Despite the warning below, this seems apromising project, where little has been done so far. To prepare for this enquiry, wesummarise some of the known properties of categories of C -algebras; we cover tensorproducts, some homotopy theory, universal properties, and triangulated structures.In addition, we examine the Universal Coefficient Theorem and the Baum–Connesassembly map.Despite many formal similarities, the homotopy theory of spaces and non-commutativetopology have a very different focus.On the one hand, most of the complexities of the stable homotopy category of spacesvanish for C -algebras because only very few homology theories for spaces have anon-commutative counterpart: any functor on C -algebras satisfying some reasonable