Homotopy Pushouts: Standard Models and Equivalence Properties

This lecture delves into the concept of homotopy pushouts, focusing on standard models like the double mapping cylinder and the mapping cylinder. The instructor explains how to identify and assemble the components to form a double mapping cylinder, providing examples to illustrate the process. The lecture also covers the importance of studying homotopy pushouts in replacing quotient constructions and understanding suspension and mapping cones. Additionally, the instructor introduces the 'strictification lemma' to ensure the commutativity of diagrams in pushout constructions, emphasizing the need for strict commutativity for universal properties. The lecture concludes with a proposition on the equivalence properties of pushouts along curve vibrations, highlighting the left properness property in topological spaces.