Left Homotopy as an Equivalence Relation: The Homotopy Relation in a Model Category

This lecture focuses on the left homotopy relation as an equivalence relation in the context of model categories. The instructor proves that the left homotopy relation is reflexive and symmetric for all objects in the model category, becoming transitive when the domain is cofibrant. The lecture delves into the properties of left homotopy, including its reflexiveness, symmetry, and transitivity under certain conditions. It also explores the concept of gluing cylinders together abstractly and demonstrates the construction of a new cylinder. The lecture concludes with a discussion on the left homotopy classes and sets the stage for the study of the set of left homotopy classes.