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Register a new userOn functors from category of Giry algebras to category of convex spaces
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Tomas Crhak
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In The factorization of the Giry monad (arXiv:1707.00488v2) the author asserts that the category of convex spaces is equivalent to the category of Eilenberg-Moore algebras over the Giry monad. Some of the statements employed in the proof of this claim have been refuted in our earlier paper (arXiv:1803.07956). Building on the results of that paper we prove that no such equivalence exists and a parallel statement is proved for the category of super convex spaces.
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The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functionals defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that states on categories as linear functionals defined on category algebras turn out to be generalized of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain representations of category algebras of $^{dagger}$-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.
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We make some beginning observations about the category $mathbb{E}mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that of $S$-equivalence classes, which is induced by a computable function. We also consider some full subcategories of $mathbb{E}mathrm{q}$, such as the category $mathbb{E}mathrm{q}(Sigma^0_1)$ of computably enumerable equivalence relations (called ceers), the category $mathbb{E}mathrm{q}(Pi^0_1)$ of co-computably enumerable equivalence relations, and the category $mathbb{E}mathrm{q}(mathrm{Dark}^*)$ whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in $mathbb{E}mathrm{q}(Sigma^0_1)$ the epimorphisms coincide with the onto morphisms, but in $mathbb{E}mathrm{q}(Pi^0_1)$ there are epimorphisms that are not onto. Moreover, $mathbb{E}mathrm{q}$, $mathbb{E}mathrm{q}(Sigma^0_1)$, and $mathbb{E}mathrm{q}(mathrm{Dark}^*)$ are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in $mathbb{E}mathrm{q}(Pi^0_1)$ whose coequalizer in $mathbb{E}mathrm{q}$ is not an object of $mathbb{E}mathrm{q}(Pi^0_1)$.
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