Weak bimonads and weak Hopf monads
It was shown by Pareigis that an algebra \(A\) over a field is a bialgebra if and only if the category of (left or right) A-modules is monoidal and the forgetful functor from the category of \(A\)-modules to the category of vector spaces is strict monoidal. This was extended to bialgebras in a braided monoidal category, see \textit{S. Majid} [New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 158, 55--105 (1994; Zbl 0812.18004)], and was generalized by \textit{I. Moerdijk} to monoidal categories [J. Pure Appl. Algebra 168, No.2-3, 189--208 (2002; Zbl 0996.18005)]. He defined a bimonad on a monoidal category \(M\) as a monad \(T\) on Msuch that the Eilenberg-Moore category \(M^T\) of \(T\)-algebras is monoidal and the forgetful functor from \(M^T\) to \(M\) is is strict monoidal,i.e., the monoidal structure of M lifts to \(M^T\). Pareigis' result was generalized to weak bialgebras by \textit{K. Szlachanyi} [New York, NY: Marcel Dekker. Lecture Notes in Pure and Applied Mathematics 239, 291--307 (2005; Zbl 1064.18008)] where the condition on the forgetful functor is that it obey a separable Frobenius condition, i.e., it has both a monoidal and an opmonoidal structure which satisfy some compatiblity properties. Weak bialgebras can be defined in \(sny\) braided monoidal category, e.g., see \textit{C. Pastro} and \textit{R. Street} [Algebra Number Theory 3, No. 2, 149--207 (2009; Zbl 1185.16035)]. The paper under review generalizes weak bialgebras to monoidal categories(possibly not braided). The authors define a weak bimonad on a monoidal category \(M\) as a monad \(T\) on \(M\) with extra structure making \(M^T\) monoidal and the forgetful functor from \(M^T\) to \(M\) separable Frobenius. They establish an equivalence between the category of weak bimonads on a Cauchy complete monoidal category \(M\) and the category of bimonads on bimodule categories over separable Frobenius monoids in \(M\). THey show that weak bimonads in a braided monoidal category induce weak bimonads in certain braided momoidal categories, with a partial converse. Finally they define and study weak right(or left) Hopf monoids and antipodes.
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