A Hopf monad, in the sense of Bruguières, Lack, and Virelizier, is a special kind of monad that can be defined for any monoidal category. In this note, we study Hopf monads in the case of a category with finite biproducts, seen as a symmetric monoidal category. We show that for biproducts, a Hopf monad is precisely characterized as a monad equipped with an extra natural transformation satisfying three axioms, which we call a fusion invertor. We will also consider three special cases: representable Hopf monads, idempotent Hopf monads, and when the category also has negatives. In these cases, the fusion invertor will always be of a specific form that can be defined for any monad. Thus in these cases, checking that a monad is a Hopf monad is reduced to checking one identity.
Keywords: Hopf Monads, Biproducts, Fusion Operators, Fusion Invertor
2020 MSC: 18C15, 18M80, 18D99
Theory and Applications of Categories, Vol. 39, 2023, No. 28, pp 804-823.