Algebraic Topology: Lecture 2

We first discussed functors and natural transformations and gave examples. A good textbook is S. MacLane: Categories for the working mathematician. Second Edition, Graduate Texts in Mathematics 5, Springer-Verlag. We then introduced the notion of a model category following Quillen. 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. This means the following: There is a functor from F : C --> Ho(C) that satisfies: (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. It is not clear that the homotopy category Ho(C) exists. We will prove this in lectures 3 and 4.