Monads on dagger categories
Category Theory
2025-09-08 v3
Abstract
The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when everything 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.
Cite
@article{arxiv.1602.04324,
title = {Monads on dagger categories},
author = {Chris Heunen and Martti Karvonen},
journal= {arXiv preprint arXiv:1602.04324},
year = {2025}
}
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28 pages