process

12 R. Meyerextensions, and cp-split extensions, respectively. Similar remarks apply to mappingcylinders and mapping cones: for any morphism of extensionsI E Qwe get extensions˛ I 0 E0 Q 0 ,ˇCyl.˛/ Cyl.ˇ/ Cyl./; Cone.˛/ Cone.ˇ/ Cone./I (6)if the extensions in (5) are split or cp-split, so are the resulting extensions in (6).The familiar maps relating mapping cones and cylinders to cones and suspensionscontinue to exist in our case. For any morphism f W A ! B in G-C alg, we get amorphism of extensionsSus.B/ Cone.f / ACone.B/ Cyl.f / A.The bottom extension splits and the maps A $ Cyl.f / are inverse to each other upto homotopy. By naturality, the composite map Cone.f / ! A ! B factors throughCone.id B / Š Cone.B/ and hence is homotopic to the zero map.3 Universal functors with certain propertiesWhen we study topological invariants for C -algebras, we usually require homotopyinvariance and some exactness and stability conditions. Here we investigate theseconditions and their interplay and describe some universal functors.We discuss homotopy invariant functors on G-C alg and the homotopy categoryHo.G-C alg/ in §3.1. This is parallel to classical topology. We turn to Morita–Rieffel invariance and C -stability in §3.2–3.3. We describe the resulting localisationusing correspondences. By the way, in a C -algebra context, stability usually refers toalgebras of compact operators instead of suspensions. §3.4 deals with various exactnessconditions: split-exactness, half-exactness, and additivity.Whereas each of the above properties in itself seems rather weak, their combinationmay have striking consequences. For instance, a functor that is both C -stable andsplit-exact is automatically homotopy invariant and satisfies Bott periodicity.Throughout this section, we consider functors S ! C where S is a full subcategoryof C alg or G-C alg for some locally compact group G. The target category Cmay be arbitrary in §3.1–§3.3; to discuss exactness properties, we require C to bean exact category or at least additive. Typical choices for S are the categories ofseparable or separable nuclear C -algebras, or the subcategory of all separable nuclearG-C -algebras with an amenable (or a proper) action of G.(5)