Unifying left and right homotopy: The homotopy relation

This lecture demonstrates that left homotopic morphisms with cofibrant domain are also right homotopic, and vice versa, leading to the equivalence of right and left homotopy for morphisms between bifibrant objects. The concept of homotopy equivalence between bifibrant objects is then introduced, along with the relation between left and right homotopy. The lecture covers propositions regarding cofibrant and fibrant objects, as well as the definition of homotopy classes of morphisms. Additionally, it explores the well-defined map between homotopy classes of morphisms and the notion of homotopy equivalence. The Whitehead Lemma is presented as a concluding topic.