Category Algebras and States on Categories

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.

Quantum Fields as Category Algebras

“…The multiplication defined above is nothing but the "convolution" operation on the category C. R[C] coincides with the algebra studied in [29] if R is a ring. In [10], it is denoted as 0 R 0 [C] to distinguish them from other kinds of category algebras.…”

“…If the bijective-on-objects functor is also injective on arrows, the induced morphism becomes injective. Hence, the category algebra R In the previous work [10], we denoted the indeterminate ι c as χ c . We change the notation to avoid confusion with "character" in representation theory.…”

“…We call the states on category algebras as states on categories. If the number of objects in the category is finite, states can be characterized by functions on arrows satisfying certain conditions [10], which is a generalization of the result in [14] for groupoids with finite numbers of objects. More generally, to define a state on a category whose support is contained in a subcategory with finite numbers of objects is equivalent to defining the corresponding function which assigns the weight to each arrow.…”

“…For simplicity, in this paper, we focus on a category algebra which satisfies a suitable finiteness condition. Importantly, the category algebras can be considered as generalized matrix algebras over R as well as generalized polynomial algebras [10], which provides a platform for concrete and flexible studies as well as calculations. The extension to larger algebras is, of course, of interest but the category algebras we focus on already have rich structures as covariance and local structures of subalgebras reflect the causal and partial involution structure of the category, as we will see in Section 3.…”

“…In general, the notion of state on * -algebra over C is defined as positive normalized linear functional. We can naturally extend the notion in the context of algebras with involution over rigs ( [10] for details). We call the states on category algebras as states on categories.…”