In this talk we’ll be looking at how symmetric monoidal categories encode the many instances of monoid-like structures that appear across mathematics, and exploring different ways of universally completing them to abelian groups (in the case of honest monoids) or grouplike spaces (in the case of categories). In particular we will focus on two main examples of symmetric monoidal categories: finite sets with disjoint union, and finitely-generated projective modules over a ring with direct sum. By the end, we will see how group-completing these two categories realizes a natural relationship between stable homotopy groups and algebraic K-theory.

Group Completion of Monoids and Monoidal Categories