# Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness

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## Abstract

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## 1. Introduction

“There is no certainty in sciences where mathematics cannot be applied”(Leonardo da Vinci)

## 2. Category and Consciousness

#### 2.1. Definition of Category

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Category and Consciousness

“A neural correlate of a phenomenal family S is a neural system N such that the state of N directly correlates with the subject’s phenomenal property in S.”

#### 2.3. Categories in IIT and TTC

#### 2.3.1. Categories in IIT

#### 2.3.2. Categories in TTC

## 3. Functor, Natural Transformation, and Consciousness

#### 3.1. Definition of Functor and Natural Transformation

**Definition**

**3.**

- 1.
- It maps f: X→Y in C to F(f): F(X)→F(Y) in D;
- 2.
- F(f ∘ g) = F(f) ∘ F(g) for any (composable) pair of f and g in C;
- 3.
- For each X in C, F(1X) = 1F(X).

**Definition**

**4.**

- 1.
- t maps each object X in C to corresponding arrow tX: F(X)→G(X) in D;
- 2.
- For any f: X→Y in C, tY ∘ F(f) = G(f) ∘ tX.

#### 3.2. Functor, Natural Transformation, and Consciousness

#### 3.3. Functors and Natural Transformations in IIT and TTC

#### 3.3.1. Functors and Natural Transformations in IIT

#### 3.3.2. Functor and Natural Transformations in TTC

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Objects, arrows, domain, codomain: Each arrow f is associated with two objects, dom(f) and cod(f), which are called the domain and the codomain of (f). When dom(f) = X and cod(f) = Y, we denote f: X→Y, as shown in Figure 1a. (The direction of the arrow can be in any direction, from left to right or reverse, whichever is convenient.) A system with arrows and objects is called a diagram. (

**b**) Composition: If there are two arrows f and g, such that cod(f) = dom(g), there is a unique arrow, (

**c**) g ∘ f, called the composition of f and g. A diagram is called commutative when any compositions of arrows having the common codomain and domain are equal. (

**d**) Associative law: (h ∘ g) ∘ f = h ∘ (g ∘ f). In other words, the diagram is commutative. (

**e**) Unit law: For any object X there exists an arrow 1X: X→X, such that the diagram is commutative for any f: X→Y. In other words, f ∘ 1X = f = 1Y ∘ f for any f. 1X is called the identity of X.

**Figure 2.**(

**a**) In integrated information theory (IIT), category is defined by an object that is a stochastic network with transition probability matrix (TPM). The exemplar network is composed of a copy gate A and B, which copies the state of the other gate with a time delay of 1. The state of the gate is either on or off. The table on the right describes its TPM. An arrow in category N0 is “decomposition” of the network with TPM (Note that we grossed over various details that are important for IIT3.0 (e.g., distinction between past and future). In particular, how decomposed subnetwork should be embedded with the original network requires careful consideration of so-called “purview” in IIT3.0. Within the IIT’s algorithm, what we call “decomposition” corresponds to a step where one evaluates all potential candidate φ or small phi. For example, for a system ABC, its power set, A, B, C, AB, BC, AC, ABC needs to be evaluated. In some cases, decomposed candidate small phis may not exist, thus it may better be called as “potential decomposition”. However, for simplicity, we prefer to call it as “decomposition”.). Decomposition allows IIT to quantify the causal contribution of a part of the system to the whole. (

**b**) Disconnection arrows find the minimally disconnected network, which captures the concept of the amount of integration in IIT.

**Figure 3.**Schematic depiction of a functor: a structure-preserving mapping from one category to another category.

**Figure 4.**(

**a**) Definition of “inclusion functor”. (

**b**) Subcategory C is included by category D if inclusion functor F: C->D exists. Note that C does not need to be “a part of” D to be “included” (unlike a commonsense definition of “inclusion”).

**Figure 5.**(

**a**) Inclusion Functor i: N0→N1. N0 is included in N1 through Inclusion Functor i. (

**b**) Expansion Functor e: N0→N1. e is a different structure preserving mapping from N0 to N1 (i.e., a functor from N0 to N1), but there is “natural transformation” from i to e.

**Figure 6.**Schematic depiction of a natural transformation: a structure-preserving mapping from one functor to another functor.

**Figure 7.**(

**a**) Inclusion functor, i, expansion functor, e, in the IIT category N0 (actual) and N1 (all possible). Objects in N0 and N1 (e.g., [AB]) are a network with TPM, and arrows in N0 and N1 are manipulation of network/TPM that is allowed in IIT. Within N0, we consider only decomposition arrows. N1 is enriched by additional disconnection arrows that represent an operation that finds a “minimally disconnected” network with TPM within N1. An expansion functor, e, finds the minimally disconnected network (e.g., [AB]’) of the original network (e.g., [AB]), as well e also preserves the structure of N0, and qualifies as a functor. A red arrow within N1 that goes from the actual to the minimally disconnected network corresponds to integrated information, φ. (

**b**) Considering decomposition arrows in N0 allows N0 to consist of a powerset of the network. If natural transformation, t, from the inclusion to the expansion functor exists, t gives us a power set of φ’s, the original and the minimally disconnected network with TPMs. This corresponds to system level integration, Φ.

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**MDPI and ACS Style**

Northoff, G.; Tsuchiya, N.; Saigo, H. Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness. *Entropy* **2019**, *21*, 1234.
https://doi.org/10.3390/e21121234

**AMA Style**

Northoff G, Tsuchiya N, Saigo H. Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness. *Entropy*. 2019; 21(12):1234.
https://doi.org/10.3390/e21121234

**Chicago/Turabian Style**

Northoff, Georg, Naotsugu Tsuchiya, and Hayato Saigo. 2019. "Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness" *Entropy* 21, no. 12: 1234.
https://doi.org/10.3390/e21121234