Traced monads and Hopf monads

JS Lemay – 24 May 2023

A traced monad is a monad on a traced monoidal category that lifts the traced monoidal structure to its Eilenberg-Moore category. A long standing question has been trying to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a Hopf monad is a comonoidal monad whose fusions operators are invertible. Furthermore, for compact closed categories, a Hopf monad is precisely the kind of monad on a compact closed category that lifts the compact closed structure to its Eilenberg-Moore category. Since every compact closed category is also a traced monoidal category, a natural question to ask is what is the relationship between Hopf monads and traced monads. In this talk, I will discuss the trace-coherence condition for Hopf monads, which can be stated without mentioning the Eilenberg-Moore category. The main result being that a Hopf monad is traced if and only if it satisfies the trace-coherence condition. I will also discuss examples of traced/Hopf monads that are not Hopf/traced monads.

This is joint work with Masahito Hasegawa. Paper available here: