On Hopf monoids in duoidal categories. (English) Zbl 1323.16023
A duoidal category \(M\) is a category with two monoidal products, with certain compatibilities. Aguiar and Mahajan defined the notion of bimonoids in such a category. As there is in general no convolution product in \(M\), there is a problem to define the antipode, and hence the notion of Hopf monoid in \(M\).
An answer is given here by the fundamental theorem of Hopf modules: if \(A\) is a bialgebra over a commutative field \(k\), the following assertions are equivalent: 1. \(A\) has an antipode. 2. \(A\) is an \(A\)-Galois extension of \(k\). 3. (Fundamental theorem of Hopf modules). The category of \(A\)-Hopf modules is equivalent to the category of \(k\)-modules.
A similar theorem is proved under certain conditions on \(M\), namely the Fundamental Theorem of Hopf modules holds for a bimonoid \(A\) if, and only if, the unit of \(A\) determines an \(A\)-Galois extension. These results are applied to small groupoids and to Hopf algebroids over a commutative base algebra.
An answer is given here by the fundamental theorem of Hopf modules: if \(A\) is a bialgebra over a commutative field \(k\), the following assertions are equivalent: 1. \(A\) has an antipode. 2. \(A\) is an \(A\)-Galois extension of \(k\). 3. (Fundamental theorem of Hopf modules). The category of \(A\)-Hopf modules is equivalent to the category of \(k\)-modules.
A similar theorem is proved under certain conditions on \(M\), namely the Fundamental Theorem of Hopf modules holds for a bimonoid \(A\) if, and only if, the unit of \(A\) determines an \(A\)-Galois extension. These results are applied to small groupoids and to Hopf algebroids over a commutative base algebra.
Reviewer: Loïc Foissy (Calais)
MSC:
16T05 | Hopf algebras and their applications |
18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
16T10 | Bialgebras |
18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
18C20 | Eilenberg-Moore and Kleisli constructions for monads |