The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleisli and Frobenius-Eilenberg-Moore algebras, which again have a dagger. We characterize the Frobenius law as a coherence property between dagger and closure, and characterize strong such monads as being induced by Frobenius monoids.
Keywords: Dagger category, Frobenius monad, Kleisli algebra, Eilenberg-Moore algebra
2010 MSC: 18A40, 18C15, 18C20, 18D10, 18D15, 18D35
Theory and Applications of Categories, Vol. 31, 2016, No. 35, pp 1016-1043.