Lecture
This lecture discusses the fundamental concept of lifting properties in model categories, focusing on left and right lifting properties. The instructor begins by introducing the significance of these properties in the context of homotopy theory and model categories. The discussion includes the definition of left lifting properties and their implications for morphisms within a category. The instructor emphasizes the importance of commutative diagrams and how lifting properties relate to them. The lecture also covers the preservation of lifting properties under various operations, such as push-outs and pull-backs. Additionally, the instructor explores the relationship between lifting properties and retracts, providing examples to illustrate these concepts. The session concludes with a discussion on the implications of lifting properties in the category of sets, highlighting the conditions under which morphisms possess left lifting properties. Overall, the lecture provides a comprehensive overview of lifting properties and their relevance in the study of model categories.