Algebraic Topology: Lecture 5

We introduced the notion of a model category following Quillen's paper. A model category consists of a category C together with three classes of maps called weak equivalences, fibrations, and cofibrations. The associated homotopy category Ho(C) is obtained from C by formally inverting the weak equivalences. More precisely, there is a functor from F : C --> Ho(C) with the following properties: (i) If f is a weak equivalence in C then F(f) is an isomorphism in Ho(C). (ii) If G : C --> D is a functor such that G(f) is an isomorphism in D whenever f is a weak equivalence in C, then there exists a unique functor H : Ho(C) --> D such that G = H o F. We began the proof that the homotopy category exists, but we did not finish the proof today. For reading see Paragraph 1 of Chapter 1 of Quillen's paper.