Existence of Left Derived Functors

This lecture discusses the existence of left derived functors in the context of homotopical algebra. It covers the conditions under which a functor admits a left derived functor, focusing on weak equivalences between cofibrant and fibrant objects. The lecture also delves into the construction of induced functors and required natural transformations. Through a step-by-step analysis, the instructor demonstrates the process of constructing left derived functors and their properties, emphasizing the isomorphism conditions for different types of objects. The lecture concludes with a detailed explanation of how the functor preserves composition and the well-definedness of the constructed functors on morphisms.