Traced Monads and Hopf Monads

Abstract

A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.