Extensive categories, commutative semirings and Galois theory

We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B

Frobenius-Eilenberg-Moore objects in dagger 2-categories

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.Non UBCUnreviewedAuthor affiliation: University of Cape TownGraduat