Algebraic Topology: Lecture 13

We discussed the following two results: A Quillen adjunction (F, G, a) between two model categories C and D gives rise to an adjunction (LF, RG, Ra) between the homotopy categories Ho(C) and Ho(D). We call this the derived adjunction. The Quillen adjunction (F, G, a) is called a Quillen equivalence if for every cofibrant object X of C and every fibrant object Y of D, a map f : F(X) --> Y is a weak equivalence in D if and only if the adjoint map a(f) : X --> G(Y) is a weak equivalence in C. In this case the derived adjunction (LF, RG, Ra) is an adjoint equivalence between the categories Ho(C) and Ho(D). This means that a map f : (LF)(X) --> Y is an isomorphism in Ho(D) if and only if (Ra)(f) : X --> (RG)(Y) is an isomorphism in Ho(C).

We then began a discussion of simplicial sets and of the adjunction (| - |, Sin, a) between the categories of simplicial sets and topological spaces given by the geometric realization functor and the singular set functor. We mentioned that there exists a model structure on the category of simplicial sets such that this adjunction is a Quillen equivalence. Hence, for the purpose of doing homotopy theory, there is no loss of information by considering simplicial sets instead of topological spaces.