Derived functors play a central role in commutative algebra, organizing algebraic computations through exact sequences and homological methods. This talk begins with a brief conceptual revisit of derived functors in homological algebra, and then examines Kähler differentials of commutative rings as a motivating example for why we want derived functors beyond abelian categories.

To address this problem, we trace Dan Quillen’s insight that homotopical methods provide an ideal framework for defining derived functors in nonabelian settings. Along the way, we encounter his definition of model categories, which have since become central tools in modern homotopy theory. From this perspective, the cotangent complex arises as a derived replacement for Kähler differentials. We will outline the construction of the cotangent complex and discuss some of its powerful applications to deformation theory. Time permitting, I will say something about how the homotopical viewpoint reconciles with homological algebra.