Algebraic Topology: Lecture 4

Let C be a model category. We proved the following techincal lemmas: 1) If A is a cofibrant object, and if Cyl(A) is a cylinder object for A, then the maps d^0, d^1: A --> Cyl(A) are both cofibrations and weak equivalences. 2) If A is cofibrant, then the relation of left homotopy on Hom(A,B) is an equivalence relation. 3) Let f, g: A --> B be left homotopic maps, and suppose that A is cofibrant. Then f and g are also right homotopic. 4) Let f, g: A --> B and u: B --> C be maps and suppose that A is cofibrant. If f and g are right homotopic then so are uf and ug. 5) Hom(A,B)/~r denote the set of equivalence classes for the equivalence relation on the set Hom(A,B) generated by the relation of right homotopy. Suppose that A is cofibrant. Then the composition in C induces a map Hom(B,C)/~r x Hom(A,B)/~r --> Hom(A,C)/~r. 6) Let pi^l(A,B) denote the set of equivalence classes for the equivalence relation on the set Hom(A,B) generated by the relation of left homotopy. Let A be a cofibrant object, and let p: X --> Y be a map that is both a fibration and a weak equivalence. Then p induces a map p_*: Hom(A,X)/~l --> Hom(A,Y)/~l and this map is a bijection.

We shall use these technical results next time to show that the homotopy category Ho(C) exists.