Homotopy Theory: Cylinders and Path Objects

This lecture discusses the concepts of cylinders, path objects, and homotopy in the context of model categories. The instructor begins by recalling definitions related to model categories, including left and right homotopy. The lecture emphasizes the importance of these concepts in homotopical algebra, particularly in understanding morphisms between objects. The instructor provides examples to illustrate how cylinders and path objects function within the category of sets and topological spaces. The discussion includes conditions under which left homotopy is equivalent to right homotopy, and the implications of these relationships. The instructor also explores the significance of very good cylinders and their role in establishing homotopies. The lecture concludes with a transition to more complex examples, particularly in the context of topological spaces, highlighting the differences and similarities in homotopical reasoning across various categories. Overall, the lecture aims to deepen the understanding of homotopy theory and its applications in mathematics.