Category Algebras and States on Categories

TL;DR

This paper develops a novel connection between category theory and noncommutative probability by introducing states on category algebras, generalizing probability measures, and extending the GNS construction to semi-Hilbert modules.

Contribution

It introduces the concept of states on category algebras and generalizes the GNS construction within a categorical framework, bridging category theory and quantum probability.

Findings

Category algebras can be viewed as generalized matrix algebras.

States on categories generalize probability measures on sets.

Representation of category algebras on semi-Hilbert modules is established.

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 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.