Derived Functors in Homotopical Algebra

This lecture focuses on proving the Fundamental Theorem of homotopical algebra, which establishes that a Quillen pair induces an adjunction of the corresponding homotopy categories, while a Quillen equivalence induces an equivalence of the homotopy categories. The instructor explains the construction of natural transformations and the analysis involved in the proof. The lecture also covers the concept of derived functors, Quillen pairs, and Quillen equivalences in the context of model categories. The process of defining natural transformations and ensuring well-definedness of mappings is discussed, emphasizing the importance of homotopy invariance. The lecture concludes by demonstrating the implications of the Fundamental Theorem in the context of homotopical algebra.