process

Categorical aspects of bivariant K-theory 13Definition 26. Let P be a property for functors defined on S. A universal functorwith P is a functor uW S ! Univ P .S/ such that• xF ı u has P for each functor xF W Univ P .S/ ! C;• any functor F W S ! C with P factors uniquely as F D xF ı u for some functorxF W Univ P .S/ ! C.Of course, a universal functor with P need not exist. If it does, then it restricts toa bijection between objects of S and Univ P .S/. Hence we can completely describeit by the sets of morphisms Univ P .A; B/ from A to B in Univ P .S/ and the mapsS.A; B/ ! Univ P .A; B/ for A; B 22 S; the universal property means that forany functor F W S ! C with P there is a unique functorial way to extend the mapsHom G .A; B/ ! C F .A/; F .B/ to Univ P .A; B/. There is no a priori reason whythe morphism spaces Univ P .A; B/ for A; B 22 S should be independent of S; butthis happens in the cases we consider, under some assumption on S.3.1 Homotopy invariance. The following discussion applies to any full subcategoryS G-C alg that is closed under the cylinder functor.Definition 27. Let f 0 ;f 1 W A B be two parallel morphisms in S. We write f 0 f 1and call f 0 and f 1 homotopic if there is a homotopy between f 0 and f 1 , that is, amorphism f W A ! Cyl.B/ D C.Œ0; 1; B/ with ev t ı f D f t for t D 0; 1.It is easy to check that homotopy is an equivalence relation on Hom G .A; B/. We letŒA; B be the set of equivalence classes. The composition of morphisms in S descendsto mapsŒB; C ŒA; B ! ŒA; C ; Œf ; Œg 7! Œf ı g;that is, f 1 f 2 and g 1 g 2 implies f 1 ı f 2 g 1 ı g 2 . Thus the sets ŒA; B formthe morphism sets of a category, called homotopy category of S and denoted Ho.S/.The identity maps on objects and the canonical maps on morphisms define a canonicalfunctor S ! Ho.S/. A morphism in S is called a homotopy equivalence if it becomesinvertible in Ho.S/.Lemma 28. The following are equivalent for a functor F W S ! C:(a) F.ev 0 / D F.ev 1 / as maps F C.Œ0; 1; A/ ! F .A/ for all A 22 S;(b) F.ev t / induces isomorphisms F C.Œ0; 1; A/ ! F .A/ for all A 22 S, t 2 Œ0; 1;(c) the embedding as constant functions const W A ! C.Œ0; 1; A/ induces an isomorphismF .A/ ! F C.Œ0; 1; A/ for all A 22 S;(d) F maps homotopy equivalences to isomorphisms;(e) if two parallel morphisms f 0 ;f 1 W A B are homotopic, then F.f 0 / D F.f 1 /;(f) F factors through the canonical functor S ! Ho.S/.Furthermore, the factorisation Ho.S/ ! C in (f) is necessarily unique.