Explore polynomial functors in this Topos Institute Colloquium talk that delves into their evolution from basic categorification of polynomial functions to powerful tools for handling induction, nesting, and substitution. Learn how endofunctors of the category of sets, built from sums and products, serve as fundamental building blocks in combinatorics, logic, and homotopy theory. Discover the theoretical framework that bridges elementary arithmetic concepts with complex mathematical structures like infinity-operads through detailed explanations and practical applications.