**1. Introduction**

In the present paper, we study category algebras and states defined on arbitrary small categories to build a new bridge between category theory (see [1–4] and references therein, for example) and noncommutative probability or quantum probability (see [5–7] and references therein, for example), a generalized probability theory which was originated as a mathematical framework for quantum theory.

A category algebra is, in short, a convolution algebra of functions on a category. For example, 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. Many authors have studied the notions of Möbius inversion, which has been one of fundamental part of combinatorics since the pioneering work by Rota [9] on posets, in the context of incidence algebras on categories ([8,10–14], for example).

There is another approach to obtain the notion of category algebra. As is well known, a group algebra is defined as a convolution algebra consisting of finite linear combinations of elements. By generalization with replacing "elements" by "arrows", one can obtain another notion of category algebra (see [13], for example), which also includes monoid algebra (in particular polynomial algebras) and groupoid algebras as examples. Please note that for a category with infinite number of objects, the algebra is not unital.

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]. These category algebras include the ones studied in [13] as subalgebras in general, and they coincide for categories with finite number of objects. Moreover, one of the algebras we study, called "backward finite category algebra", coincides with incidence algebras for combinatorically important cases originally studied in [9].

**Citation:** Saigo, H. Category Algebras and States on Categories. *Symmetry* **2021**, *13*, 1172. https:// doi.org/10.3390/sym13071172

Academic Editors: Motoichi Ohtsu and Alexey Kanel-Belov

Received: 28 May 2021 Accepted: 26 June 2021 Published: 29 June 2021

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The purpose of this paper is to provide a new framework for the interplay between regions of mathematical sciences such as algebra, probability and physics, in terms of states as linear functional defined on category algebras. As is well known, quantum theory can be considered to be a noncommutative generalization of probability theory. At the beginning of quantum theory, matrix algebras played a crucial role (see [16] for example). In the present paper, 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]).

Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction [18,19] (as for the studies in category theoretic context, see [20–22] for example), we obtain a representation of category algebras of †-categories on certain generalized 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.

**Notation 1.** *In the present paper, categories are always supposed to be small (This assumption may be relaxed by applying some appropriate foundational framework). The set of all arrows in a category* C *is also denoted as* C*.* |C| *denotes the set of all objects, which are identified with corresponding identity arrows, in* C*. We also use the following notations:*

$$\prescript{}{}{\mathcal{C}}'\_{\mathbb{C}} := \mathcal{C}(\mathbb{C}, \mathbb{C}'), \; \mathcal{C}\_{\mathbb{C}} := \sqcup\_{\mathbb{C}' \in [\mathcal{C}]} \mathcal{C}(\mathbb{C}, \mathbb{C}'), \; \prescript{}{\mathcal{C}}' \mathcal{C} := \sqcup\_{\mathbb{C} \in [\mathcal{C}]} \mathcal{C}(\mathbb{C}, \mathbb{C}'),$$

*where* C(*<sup>C</sup>*, *C* ) *denotes the set of all arrows from C to C .*
