Model categories with a view towards rational homotopy theory

This thesis is an investigation into the theory of model categories with applications in the foundations of rational homotopy theory. For a model category C we construct the associated homotopy category Ho(C) and prove that this construction yields a localization of C with respect to the class of weak equivalences. We also prove a total derived functor theorem. This gives sufficient conditions for a functor between two model categories to induce an equivalence of categories between the associated homotopy categories.

We then turn our attention to some specific examples of model categories. We put a model structure on the category of r-reduced simplicial sets and r-reduced simplicial groups. In both cases the weak equivalences are those maps inducing isomorphism on rational homotopy groups. We then prove a general result which, given a category C (satisfying some conditions), provides a model structure on the category sC of simplicial objects in C. This theorem is applied to give model structures on the category of r-reduced simplicial complete rational Hopf algebras and the r-reduced simplicial rational Lie algebras, respectively. Finally we put a model structure on the category of r-reduced differential graded rational Lie algebras, where the weak equivalences are the maps inducing isomorphism on homology. As we prove the model category axioms for the various structures we also construct pairs of adjoint functors between them and show that these satisfy the conditions of the total derived functor theorem. As a result, the homotopy category Ho(sSet_1^Q) of 1-reduced simplicial sets is equivalent to the homotopy category Ho(dgLie_0) of connected differential graded Lie algebras.