Introduction to Derived Functors: Left and Right Derived Functors

This lecture introduces the concepts of left and right derived functors in homotopical algebra, inspired by the notion of derived functors in homological algebra. The instructor explains how any functor between model categories that sends weak equivalences to isomorphisms induces both left and right derived functors. The lecture covers the definition and properties of left and right derived functors, emphasizing their uniqueness up to unique natural isomorphism. An example is provided to illustrate that if a functor sends weak equivalences to isomorphisms, then it admits both left and right derived functors, which are essentially the same.