Quillen pairs and Quillen equivalences: Derived functors

This lecture introduces conditions on an adjunction between model categories that ensure the existence of an adjunction between the associated homotopy categories, given by the total left derived functor of the left adjoint and the total right derived functor of the right adjoint. Further conditions are provided under which this adjunction becomes an equivalence of categories. The lecture covers the definition of Quillen pairs and Quillen equivalences, exploring the conditions for LF or RF to be an adjoint. It also discusses the conditions for a Quillen equivalence, emphasizing the preservation of weak equivalences. The lecture concludes with the fundamental theorem of homotopical algebra, stating the properties of LF and RG in the context of Quillen pairs and equivalences.