*2.2. Relativistic Structure as Category*

If we intuitively consider the spacetime degrees of freedom with the geometric notion of the "set" of possible events, it is natural to think that the structure of dynamics of quantum can be modeled with "category" as the total system structure of relationships.

**Definition 16** (causal category)**.** *A category* C *equipped with a subcategory* C*cau satisfying* |C| = |C*cau*| *is called a causal category. Arrows in* C*cau are said to be causal.*

Any category can be considered as a causal category by taking C = C*cau*. Note that |C| is equipped with preorder , defined as the existence of causal arrows between objects.

A typical example of causal categories is constructed as follows. For a spacetime (with inner degrees of freedom) *E*, usually modeled by a manifold and sometimes by a symmetric directed graph (as in the lattice gauge theory [27]), we can construct a category C = M[*E*] whose objects and arrows are points and paths between them. More precisely, we consider M(*E*) as a subcategory of the "Moore path category" [28] of *E*, consisting of smooth paths in the manifold case and as the free category of *E* in the discrete case. Then, we can define C*cau* as the subcategory consisting of "causal paths". For the manifold case, the notion of causal paths can be defined as the paths whose tangent vectors are all in the future light cone. For the graph case, a path (i.e., an arrow in the free category) *c* is said to be causal if *c* = *c* ◦ *c* implies dom(*c* ) *cod*(*c* ) and dom(*c*) *cod*(*c*), where denotes a preorder previously defined on the set of vertices.

**Definition 17** (relevant category)**.** *Let* C *be a causal category and* O *be a subset of* |C|*. The subcategory of* C *generated by*

*arrows whose domain and codomain are in* O*; causal arrows whose domain is in* O *and whose codomain is in* |C| \ O*; causal arrows whose codomain is in* O *and whose domain is in* |C| \ O*; and identity arrows (identified with objects) in* |C| \ O*,*

*is called the relevant category for* O *and denoted as* O*rel.*

> By the definition of relevant categories, the following structure theorem holds.

**Theorem 1** (structure theorem for relevant category)**.** *Let* C *be a causal category and* O *be a subset of* |C|*. Any arrow in the relevant category* O*rel can be written in either of the following forms:*

$$
\mathcal{C}\_{\epsilon} \mathcal{c}^{\text{out}} \circ \mathcal{c}\_{\epsilon} \mathcal{c} \circ \mathcal{c}^{\text{in}}{}\_{\epsilon} \mathcal{c}^{\text{out}} \circ \mathcal{c} \circ \mathcal{c}^{\text{in}}{}\_{\epsilon} \mathcal{i}\_{\epsilon}
$$

*where c denotes an arrow whose domain and codomain is in* O*; cout denotes a causal arrow whose domain is in* O *and whose codomain is in* |C| \ O*; cin denotes a causal arrow whose codomain is in* O *and whose domain is in* |C| \ O*; and i denotes an identity arrow in* |C| \ O*.*

The notion below is quite important to see the essence of the relativistic structure.

**Definition 18** (spacelike separated)**.** *Let* C *be a causal category and* O, O *be a subset of* |C|*.* O *and* O *are said to be spacelike separated if there is no causal arrow between their objects.*

By definition, two spacelike separated subsets are disjoint considering the identity arrows are causal. Moreover, we have the following directly from the structure theorem of the relevant category.

**Theorem 2** (non-existence of non-trivial compositable pair)**.** *Let* C *be a causal category and* O, O *be a pair of spacelike separated subsets of* |C|*. There is no pair of arrows* (*c*, *c* ) ∈ |C| × |C| / *satisfying c* ∈ O*rel, c* ∈ (O )*rel and* cod(*c*) = dom(*c* )*.*

For the application to quantum theory, the involution structure is important. From now on, we consider a causal category with partial involution structures as defined below.

**Definition 19** (partial involution structure on category)**.** *Let* C *be a category. A partial involution structure on* C *is a subcategory* C∼ *equipped with an involution such that* |C| = |C∼|*.*

Note that any category C has the trivial partial involution structure, since C is equipped with the involution structure |C|, defined as *C*† = *C*.

The notion is important because the category C∼ physically means the category consisting of "bidirectional" processes. Although this notion is a generalization of the core (i.e., the maximal groupoid in a category consisting of isomorphisms), it does not require the reversibility of the process in the meaning of invertible arrows as isomorphisms. The author believes that this generalization from groupoids to categories and from cores to partial involution structures is quite important for the application to physical phenomena, which include irreversibility.

Based on the partial involution structure, we define the notion of relevant category with involution.

**Definition 20** (relevant category with involution)**.** *Let* C *be a causal category with the partial involution structure* C∼*. The maximal subcategory* O*rel*∼ *of* C∼ *closed under the involution is called the relevant category with involution on* O*.*

The importance of the relevant categories with involution concerns the fact that we can naturally define algebras with involution from them. We will see the details of this in the next section.

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

In the previous section, we introduced the notion of causal category equipped with partial involution structures as a generalized "relativity" structure. To combine this structure with the "quantum" structure, which can be modeled by noncommutative algebras, especially effectively by noncommutative algebras over C as history has shown, we need noncommutative algebras that reflect the structures of categories: category algebra is just the right concept.
