Title: When is a Hopf Monad a Trace Monad?

Abstract:

Trace monads are monads on trace monoidal categories which lift the trace to the Eilenberg-Moore category. Hopf Monads are comonoidal monads whose fusion operators are invertible, and it has been shown that a Hopf monad on a compact closed category lifts the compact closed structure to the Eilenberg-Moore category. Since every compact closed category is also a trace monoidal category, a natural question to ask is what is the relationship between Hopf monads and trace monads. In this talk, I will give introductions to trace monads and Hopf monads, and give a necessary condition for when a Hopf monad is a trace monad, and also give examples of trace monads that are not Hopf monads. We conjecture that not all Hopf monads are a trace monad, but unfortunately, we do not yet have an example of a Hopf monad which is not a trace monad! This is joint work with Masahito Hasegawa.