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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.
@article{TAC_2023_39_a27,
author = {Masahito Hasegawa and Jean-Simon Pacaud Lemay},
title = {Hopf {Monads} on {Biproducts}},
journal = {Theory and applications of categories},
pages = {804--823},
publisher = {mathdoc},
volume = {39},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2023_39_a27/}
}
Masahito Hasegawa; Jean-Simon Pacaud Lemay. Hopf Monads on Biproducts. Theory and applications of categories, Tome 39 (2023), pp. 804-823. http://geodesic.mathdoc.fr/item/TAC_2023_39_a27/