Symmetric power structures on algebraic K-theory
This is a talk about the lambda-operations and their relatives on algebraic K-theory, which we argue should be thought of as analogous to power operations in a homology theory. Classically constructed as operations on homotopy groups, we extend the lambda-operations to the space level by using the surprising observation that polynomial functors between additive categories give maps of K-theory spaces. Subsequently, we use the language of Lawvere theories to define and study a category of spectral lambda-rings. We end with some speculations concerning how these structures might relate to the trace map. This is joint work in progress with Barwick, Mathew and Nikolaus.