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Frobenius-Eilenberg-Moore objects in dagger 2-categories

Banff International Research Station for Mathematical Innovation and Discovery

A Frobenius monad on a category is a monad-comonad pair whose multiplication and comultiplication are related via the Frobenius law. Street has given several equivalent definitions of Frobenius monads. In particular, they are those monads induced from ambidextrous adjunctions. On a dagger category, much of this comes for free: every monad on a dagger category is equivalently a comonad, and all adjunctions are ambidextrous. Heunen and Karvonen call a monad on a dagger category which satisfies the Frobenius law a dagger Frobenius monad. They also define the appropriate notion of an algebra for such a monad, and show that it captures quantum measurements and aspects of reversible computing. In this talk, we will show that these definitions are exactly what is needed for a formal theory of dagger Frobenius monads, with the usual elements of Eilenberg-Moore object and completion of a 2-category under such objects having dagger counterparts. This may pave the way for characterisations of categories of Frobenius objects in dagger monoidal categories and generalisations of distributive laws of monads on dagger categories.

Mathematics; Category Theory; Homological Algebra; Algebraic Topology; Algebraic Structures

video/mp4

Author affiliation: University of Cape Town

2023-10-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86390

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/