*3.1. Category Algebra*

We introduce the notion of category algebra in this subsection, which is based on [10].

**Definition 21** (rig)**.** *A rig R is a set with two binary operations called addition and multiplication, such that*


Note that, in general, a rig can be noncommutative. The notion of center is important for noncommutative rigs.

**Definition 22** (center)**.** *A subrig Z*(*R*) *of a rig R defined as the set of elements, which are commutative with all the elements in R, is called the center of R.*

A rig *R* is commutative if and only if *Z*(*R*) = *R*. Based on the notion of rigs, we define the notion of modules and algebras over rigs.

**Definition 23** (module over rig)**.** *A commutative monoid M under addition with unit* 0 *together with a left action of R on M* (*r*, *m*) → *rm is called a left module over R if the action satisfies the following conditions:*

*1. r*(*m* + *m*) = *rm* + *rm*, (*r* + *r*)*m* = *rm* + *rm for any m*, *m* ∈ *M and <sup>r</sup>*,*r* ∈ *R; and 2.* 0*m* = 0, *r*0 = 0 *for any m* ∈ *M and r* ∈ *R.*

*Dually, we can define the notion of right module over R. Let M be the left and right module over R. M is called an R-bimodule if*

$$r'(mr) = (r'm)r$$

*holds for any <sup>r</sup>*,*r* ∈ *R and m* ∈ *M. The left/right action above is called the scalar multiplication.*

**Definition 24** (algebra over rig)**.** *A bimodule A over R is called an algebra over R if it is also a rig with respect to its own multiplication, which is compatible with scalar multiplication, i.e.,*

$$(r'a')(ar) = r'(a'a)r, \; (a'r)a = a'(ra)r$$

*for any a*, *a* ∈ *A and <sup>r</sup>*,*r* ∈ *R.*

> We define the principal notion of the present paper:

**Definition 25** (category algebra)**.** *Let* C *be a category and R be a rig. An R-valued function α defined on* C *is said to be of finite propagation if for any object C there are, at most, a finite number of arrows whose codomain or domain is C. The module over R consisting of all R-valued functions of finite propagation together with the multiplication defined by*

$$(\alpha'\alpha)(\mathfrak{c}^{\prime\prime}) = \sum\_{\{(\mathfrak{c}^{\prime},\mathfrak{c}) \mid \mathfrak{c}^{\prime\prime} = \mathfrak{c}^{\prime} \text{cc}\}} \mathfrak{a}^{\prime}(\mathfrak{c}^{\prime})\mathfrak{a}(\mathfrak{c}), \ \mathfrak{c}, \mathfrak{c}^{\prime}, \mathfrak{c}^{\prime\prime} \in \mathcal{C}$$

*becomes an algebra over R with unit . This is defined by*

$$\epsilon(c) = \begin{cases} 1 & (c \in |\mathcal{C}|) \\ 0 & (otherwise) \end{cases}$$

*and is called the category algebra of finite propagation, which is denoted as <sup>R</sup>*[C]*. In the present paper, we simply call R*[C] *the category algebra of* C*.*

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 <sup>0</sup>*R*0[C] to distinguish them from other kinds of category algebras.

A functor from one category to another induces a homomorphism between the corresponding category algebras if the functor is bijective on objects. If the bijective-on-objects functor is also injective on arrows, the induced morphism becomes injective. Hence, the category algebra *R*[C◦] of a subcategory C◦ of a category C becomes a subalgebra of *<sup>R</sup>*[C].

**Definition 26** (indeterminate)**.** *Let R*[C] *be a category algebra and c* ∈ C*. The function ιc* ∈ *R*[C] *defined as*

$$\iota^{\varepsilon}(\mathfrak{c}') = \begin{cases} 1 & (\mathfrak{c}' = \mathfrak{c}) \\ 0 & (otherwise) \end{cases}$$

*is called the indeterminate corresponding to c.*

In the previous work [10], we denoted the indeterminate *ιc* as *χ<sup>c</sup>*. We change the notation to avoid confusion with "character" in representation theory.

For indeterminates, it is easy to obtain the following.

**Theorem 3** (calculus of indeterminates)**.** *Let c*, *c* ∈ C*, ιc*, *ιc be the corresponding indeterminates and r* ∈ *R. Then,*

$$\mu^{\varepsilon'} \iota^{\varepsilon} = \begin{cases} \iota^{\varepsilon' \odot \varepsilon} & (\text{dom}(\mathbf{c}') = \text{cod}(\mathbf{c})), \\ 0 & (\text{otherwise}), \end{cases}$$

$$\iota \iota^{\varepsilon} = \iota^{\varepsilon} r.$$

In short, a category algebra *R*[C] is an algebra of functions on C, equipped with the multiplication which reflects the compositionality structure of C. By the identification of *c* ∈ C → *ιc* ∈ *<sup>R</sup>*[C], categories are included in category algebras.

A category algebra can be considered as a generalized matrix algebra. In fact, matrix algebras are isomorphic to category algebras of indiscrete categories. For the basic notions and rules for matrix-like calculations in category algebras, see [10].

For the main application of the present paper, we need the involution structure on algebras.

**Definition 27** (involution on rig)**.** *Let R be a rig. An operation* (·)<sup>∗</sup> *on R preserving addition and covariant (resp. contravariant) with respect to multiplication is said to be a covariant (resp. contravariant) involution on R when* (·)<sup>∗</sup> ◦ (·)<sup>∗</sup> *is equal to the identity function on R. A rig with contravariant involution is called a* ∗*-rig.*

**Definition 28** (involution on algebra)**.** *Let A be an algebra over a rig R with a covariant (resp. contravariant) involution* (·)*. A covariant (resp. contravariant) involution* (·)<sup>∗</sup> *on A as a rig is said to be a covariant (resp. contravariant) involution on A as an algebra over R if it is compatible with scalar multiplication, i.e.,*

> (*rar*)<sup>∗</sup> = *ra*<sup>∗</sup>*r (covariant case)*, (*rar*)<sup>∗</sup> = *ra*<sup>∗</sup>*r (contravariant case)*.

*An algebra A over a* ∗*-rig R with contravariant involution is called a* ∗*-algebra over R.*

**Theorem 4** (category algebra as algebra with involution)**.** *Let* C *be a category with a covariant (resp. contravariant) involution* (·)† *and R be a rig with a covariant (resp. contravariant) involution* (·)*. Then, the category algebra R*[C] *becomes an algebra with covariant involution (resp.* ∗*-algebra) over R.*

### *3.2. Quantum Fields as Category Algebras*

In this section, we will show that category algebras provide appropriate models for quantum fields. As already mentioned in the introduction, a quantum field is intuitively a synthesis of relativistic and quantum structures. In the previous section, we argued that the relativistic structure as the basic structure of possible dynamics can be understood from a general point of view by the causal category. The next problem is to construct a noncommutative algebra which is consistent with relativistic covariance as well as with causality. The category algebra *<sup>R</sup>*[C], where *C* is a causal category equipped with partial involution structure, is just such an algebra.

Note that by generalizing groupoid algebras to category algebras, we can naturally incorporate processes that are not necessarily reversible. If we focus on the core of the category, i.e., the subcategory consisting of all invertible arrows, we have the corresponding groupoid algebra, which is a subalgebra of *<sup>R</sup>*[C].

**Definition 29** (quantum field)**.** *Let* C *be a causal category with partial involution structure* C∼ *and R be a rig with involution. The category algebra R*[C] *is called the quantum field on* C *over R.*

For quantum physics, the cases in which *R* is some ∗-algebra over C are important. The category C is considered as "spacetime with inner degrees of freedom of the field". Note that a quantum field on a causal category C over a rig *R* might be isomorphic to or embedded into another quantum field on another causal category C over another *R*. Hence, even if we focus on the case that *R* = C, we might cover many kinds of quantum fields. Nevertheless, we maintain letting *R* be a general rig *R* with involution when we can in the present paper for future applications.

Let us see how a quantum field, as a category algebra, incorporates the relativistic covariance structure. To begin, let us assume that a group *G* (say, the Poincaré group) acts on |C| and there is a map *u*(·) sending a pair (*g*, *C*) ∈ *G* × C to the arrow *<sup>u</sup>*(*g*,*<sup>C</sup>*) : *C* −→ *gC* in C, satisfying *<sup>u</sup>*(*gg*,*<sup>C</sup>*) = *<sup>u</sup>*(*g*,*g<sup>C</sup>*) ◦ *<sup>u</sup>*(*g*,*<sup>C</sup>*) and *<sup>u</sup>*(*<sup>e</sup>*,*<sup>C</sup>*) = *C*, where *e* denotes the unit of *G* and *C* denotes the identity arrow on *C* in the last equation. Note that each *<sup>u</sup>*(*g*,*<sup>C</sup>*) is an invertible arrow. Then, we can define the endfunctor *u* # *g* : C −→ C by

$$\widehat{u^{\varsigma}}(\mathfrak{c}) = u^{(\mathfrak{g}, \text{cod}(\mathfrak{c}))} \circ \mathfrak{c} \circ (u^{(\mathfrak{g}, \text{dom}(\mathfrak{c}))})^{-1}$$

which becomes invertible and induces the corresponding isomorphism on the category algebra *<sup>R</sup>*[C]. Note also that *<sup>u</sup>*(*g*,·) becomes a natural equivalence from C (identity functor on C) to *u* # *g*.

In general, given a natural equivalence *u* from the identity functor C to an invertible functor *u*ˆ from C to C, we can define an invertible element *ιu* ∈ *R*[C] as

$$\iota^u(c) = \begin{cases} 1 & (c \text{ is a component of } u) \\ 0 & (\text{otherwise}), \end{cases}$$

and isomorphism *ι u* on *R*[C] as

$$
\widetilde{\iota}^{\widetilde{\mu}}(a) = \iota^{\mu} a(\iota^{\mu})^{-1}.
$$

This kind of transformation will be useful to study flows, generators, and symmetries such as the local gauge invariance from the viewpoint of category algebras. In sum, the category algebra intrinsically incorporates covariance structures coherent with the structure of "spacetime" category C.

In order to consider the essential features of relativity, it is necessary to consider the structure of causal categories. For this purpose, let us consider the category algebras on relevant categories and relevant categories with involution.

**Definition 30** (relevant algebra and local algebra)**.** *Let R be a rig and* C *be a causal category. The category algebra R*[O*rel*] *is called the relevant algebra on* O *over R. The subrig Rloc*[O] *of R*[O*rel*] *whose elements are in the form of α* + *δ, where α denotes an element in <sup>R</sup>*[O*rel*]*, satisfying α*(*C*) = 0 *for any C* ∈ |C| \ O*, and δ denotes an element in R*[O*rel*] ∩ *<sup>Z</sup>*(*R*[C])*, becomes an algebra over Z*(*R*) *and is called the local algebra on* O*.*

**Definition 31** (relevant algebra with involution and local algebra with involution)**.** *Let R be a rig with involution and* C *be a causal category with partial involution structure. The category algebra <sup>R</sup>*[O*rel*∼] *is called the relevant algebra with involution on* O *over R. The subrig <sup>R</sup>loc*∼[O] *of <sup>R</sup>*[O*rel*∼]*, whose elements are in the form of α* + *δ, where α denotes an element in <sup>R</sup>*[O*rel*∼]*,* *satisfying α*(*C*) = 0 *for any C* ∈ |C| \ O*, and δ denotes an element in <sup>R</sup>*[O*rel*∼] ∩ *<sup>Z</sup>*(*R*[C])*, becomes an algebra with involution over Z*(*R*) *and is called the local algebra with involution on* O*.*

The family of local algebras with involution {*Rloc*[O]}, especially when *R* is a ∗-algebra over C, is the counterpart of {A(O)} in AQFT [4], where A(O) denotes the observable algebra defined on the bounded region O in the spacetime. So far, our framework does not focus on the topological aspect of algebras but the conceptual correspondence between our framework and AQFT is remarkable, as we will see below.

Note that our "local" algebras in general contain a certain kind of information of the "outside" of the regions. Nevertheless, they contain no information of the local algebras corresponding to spacelike separated regions. From the structure theorem of the relevant category and the definition of local algebras, we have the following concepts.

**Theorem 5** (commutativity of spacelike separated local algebras)**.** *Local algebras Rloc*[O] *and Rloc*[O] *are commutative with each other if the regions* O *and* O *are spacelike separated from each other.*

As a collorary, we have the following.

**Theorem 6** (commutativity of spacelike separated local algebras with involution)**.** *Local algebras <sup>R</sup>loc*∼[O] *and <sup>R</sup>loc*∼[O] *with involution are commutative with each other if the regions* O *and* O *are spacelike separated from each other.*

The theorem above is the conceptual counterpart of one of the axioms called the "Einstein causality" ("Axiom E" in [4]).

### *3.3. Remarks on the Comparison to TQFT*

Our category algebraic framework of quantum field theory can also be compared to the conceptual ideas in other axiomatic approaches to quantum fields, such as Topological Quantum Field Theory (TQFT) [12,13]. In the axiomatization of TQFT, a quantum field theory is considered as a certain functor from the category of *n*-cobordism *nCob* into the category *Mod*(*R*) of modules over some unital commutative ring *R* (the typical case is *R* = C and *Mod*(*R*) = *Vect*, where *Vect* denotes the category of vector spaces over C).

We can construct such a functor in a generalized setting based on our framework. Let *C* be an object in a †-category C and *R* be a rig. We define the submodule *CR*[C] of *<sup>R</sup>*[C], consisting of elements whose support is included in the set of arrows whose codomain is *C*. Then, we can define a functor (·)*R* : C −→ *Mod*(*R*) by *cR* = *<sup>ι</sup><sup>c</sup>*(·), the multiplication of *ιc*, i.e., a module homomorphism sending each *α* ∈ dom(*c*)*R*[C] to *ιcα* ∈ cod(*c*)*R*[C], for any *c* ∈ C.

Since *nCob* is a †-category, the above construction works and we obtain a canonical functor from *nCob* to *Mod*(*R*). Although this functor does not satisfy all the technical parts of the axioms proposed in TQFT, it is coherent with the physical ideas of relativistic covariance and quantum properties behind the axioms. This coherence will become more clear after introducing the notion of states on categories in the next section.

### **4. States of Quantum Fields as States on Categories**
