The monoidal Eilenberg–Moore construction and bialgebroids

AbstractStrong monoidal functors U:C→M with left adjoints determine, in a universal way, monoids T in the category of opmonoidal endofunctors on M. Treating such opmonoidal monads as abstract “quantum groupoids” we derive Tannaka duality between right adjoint strong monoidal functors and opmonoidal monads. Bialgebroids, i.e., Takeuchi's ×R-bialgebras, appear as the special case when T has also a right adjoint. Street's 2-category of monads then leads to a natural definition of the 2-category of bialgebroids