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Authors
Hayato Saigo
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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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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.
In this paper, we consider categories with colored morphisms and functors such that morphisms assigned to morphisms with a common color have a common color. In this paper, we construct a morphism-colored functor such that any morphism-colored functor from a given small morphism-colored groupoid to any discrete morphism-colored category factors through it. We also apply the main result to a schemoid constructed from a Hamming scheme.
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field $Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $mathrm{id}_xcdot mathrm{id}_x=mathrm{id}_x$ for any object $x$. Although this condition led us to a great simplification, it was redundant and had to be dropped. Here we get rid of this condition, and provide a closed model structure in full generality. The new closed model category is as well cofibrantly generated, and it is proven to be Quillen equivalent to the closed model category $mathrm{Cat}_mathrm{dg}(Bbbk)$ of (strictly unital) dg categories over $Bbbk$, given by Tabuada [Tab1]. Dropping the condition $mathrm{id}_x^2=mathrm{id}_x$ makes the construction of the closed model structure more distant from loc.cit., and requires new constructions. One of them is a pre-triangulated hull of a wu dg category, which in turn is shown to be a wu dg category as well. One example of a weakly unital dg category which naturally appears is the bar-cobar resolution of a dg category. We supply this paper with a refinement of the classical bar-cobar resolution of a unital dg category which is strictly unital (appendix B). A similar construction can be applied to constructing a cofibrant resolution in $mathrm{Cat}_mathrm{dgwu}(Bbbk)$.
We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user friendly resource for beginners and experts alike.
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