Category Algebras and States on Categories

Hayato Saigo

doi:10.3390/sym13071172

Abstract

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 functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.

Highlights

We study category algebras and states defined on arbitrary small categories to build a new bridge between category theory and noncommutative probability or quantum probability
On certain categories called finely finite category [8], which is a categorical generalization of locally finite poset, the convolution operation can be defined on the set of arbitrary functions and it becomes a unital algebra called incidence algebra
The category algebras we focus on in the present paper are unital algebras defined on arbitrary small categories, which are slightly generalized versions of algebras studied under the name of the ring of an additive category [15]

Summary

Introduction

We study category algebras and states defined on arbitrary small categories to build a new bridge between category theory (see [1,2,3,4] and references therein, for example) and noncommutative probability or quantum probability We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of states on categories as linear functionals defined on category algebras turns out to be a conceptual generalization of probability measures on sets as discrete categories (For the case of states on groupoid algebras over the complex field C it is already studied [17]). Hilbert spaces (semi-Hilbert modules over rigs), which can be considered to be an extension of the result in [17] for groupoid algebras over C This construction will provide a basis for the interplay between category theory, noncommutative probability and other related regions such as operator algebras or quantum physics. C := tC∈|C| C(C, C 0 ), where C(C, C 0 ) denotes the set of all arrows from C to C 0

Category Algebras

Example of Category Algebras

States on Categories