  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.
