Homotopical Algebra: The Homotopy Category of a Model Category

This lecture focuses on proving that the category and functor constructed in the previous lecture indeed form a localization of the model category at its subcategory of weak equivalences, establishing that the category constructed is the homotopy category. The instructor discusses the construction of the homotopy category, emphasizing the importance of weak equivalences. Various properties of the homotopy category are explored, including the preservation of composition and the uniqueness of certain functors. The lecture concludes with a discussion on functors between homotopy categories, providing a comprehensive overview of the key concepts in homotopical algebra.