Polynomial functors and K-theory

Abstract: We will report on (long overdue) joint work with Clark Barwick, Saul Glasman and Akhil Mathew. Algebraic $K$-theory of a ring or more generally an additive category is, by its definition as a group completion, functorial in additive functors. We prove that it is in fact functorial in more functors: the so-called polynomial functors (in the sense of Eilenberg–Mac Lane) and still satisfies a universal property. This generalizes previous results by Passi, Dold and others. We will in fact show this for a stable $\infty$-category and polynomial ($= n$-excisive) functors in the sense of Goodwillie. If time permits, we will explain applications of this result for lambda-ring structures on algebraic $K$-theory and give the definition of a spectral lambda ring (i.e., a higher algebra version of a lambda ring).

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)