Hopf monads on monoidal categories
The authors define and study Hopf monads in a monoidal category \(\mathcal C\). Bimonads on \(\mathcal C\) were introduced by \textit{I. Moerdijk} [J. Pure Appl. Algebra 168, No.2-3, 189--208 (2002; Zbl 0996.18005)] as monads \(T\) on \(\mathcal C\) such that the Eilenberg-Moors category \(T^\mathcal C\) is monoidal and the forgetful functor from \(\mathcal C^T\) to \(\mathcal C\) is strict monoidal. Hopf monads on monoidal categories with duals were introduced by the first and third authors [Adv. Math. 215, No. 2, 679--733 (2007; Zbl 1168.18002)] as those \(T\) for which \(\mathcal C^T\) also has duals, and certain natural transformations play the role of left and right antipodes. To generalize to arbitrary monoidal \(\mathcal C\), left and right fusion operators are introduced as certain natural transformations, and then \(T\) is a Hopf monad if these fusion operators are invertible. They also introduce Hopf adjunctions, and show that \(T\) is a Hopf monad if and only if its adjunction is a Hopf adjunction. Hopf monads on closed monoidal categories are characterized in categorical terms, and also in terms of antipodes. Relations between Hopf algebras and Hopf monads are studied. The main results here are that if \(A\) is a bialgebra in the lax center of \(\mathcal C\), then \(A\) is a Hopf monad if and only if \(A\) is a Hopf algebra in the center of \(\mathcal C\), and that there is a categorical equivalence between central Hopf algebras in \(\mathcal C\) and augmented Hopf monads on \(\mathcal C\). Some well-known reults on Hopf algebras extend to Hopf monads, such as Maschke's semisimplicity theorem and Sweedler's structure on Hopf modules. Finally they show that a bialgebroid is a Hopf monad if and only if it is a Hopf algebroid.
scientific article