process

Categorical aspects of bivariant K-theory 11These constructions work only for pointed compact spaces. If we enlarge thecategory of C -algebras to a suitable category of projective limits of C -algebras asin [28], then we can define C 0 .X; A/ for any pointed compactly generated space X.But we lose some of the nice analytic properties of C -algebras. Therefore, I prefer tostick to the category of C -algebras itself.2.6 Cylinders, cones, and suspensions. The following definitions go back to [54],where some more results can be found. The description of homotopies above leads usto define the cylinder over a C -algebra A byCyl.A/ WD C.Œ0; 1; A/:This is compatible with the cylinder construction for spaces becauseCyl C 0 .X/ Š C Œ0; 1; C 0 .X/ Š C 0 .Œ0; 1 C ^ X/for any pointed compact space X; if we use locally compact spaces, we get Œ0; 1 Xinstead of Œ0; 1 C ^ X.The universal property of Cyl.A/ is dual to the usual one for spaces because theidentification between pointed compact spaces and commutative C -algebras is contravariant.Similarly, we may define the cone Cone.A/ and the suspension Sus.A/ byCone.A/ WD C 0 Œ0; 1 nf0g; A/;Sus.A/ WD C 0 Œ0; 1 nf0; 1g;A/Š C 0 .S 1 ; A/;where S 1 denotes the pointed 1-sphere, that is, circle. These constructions are compatiblewith the corresponding ones for spaces as well, that is,Cone C 0 .X/ Š C 0 .Œ0; 1 ^ X/;Sus C 0 .X/ Š C 0 .S 1 ^ X/:Here Œ0; 1 has the base point 0.Definition 25. Let f W A ! B be a morphism in C alg or G-C alg. The mappingcylinder Cyl.f / and the mapping cone Cone.f / of f are the limits of the diagramsMore concretely,A f ! B ev 1Cyl.B/; A f ! B ev 1Cone.B/:Cone.f / D ˚.a; b/ 2 A C 0 .0; 1; B j f.a/D b.1/ ;Cyl.f / D ˚.a; b/ 2 A C 0 .Œ0; 1; B j f.a/D b.1/ :If f W X ! Y is a morphism of pointed compact spaces, then the mapping coneand mapping cylinder of the induced -homomorphism C 0 .f /W C 0 .Y / ! C 0 .X/ agreewith C 0 Cyl.f / and C 0 Cone.f / , respectively.The cylinder, cone, and suspension functors are exact for various kinds of extensions:they map extensions, split extensions, and cp-split extensions again to extensions, split