1	Neuroscience of Consciousness, 2022, 7(2), 1–13
     2	DOI: https://doi.org/10.1093/nc/niab034
       
     3	Review Article
       
     4	Special Issue: Consciousness science and its theories
       
     5	Naotsugu Tsuchiya1,2,3,† and Hayato Saigo 4,‡
     6	1
       
     7	School of Psychological Sciences and Turner Institute for Brain and Mental Health, Monash University, Melbourne, VIC 3800, Australia; 2 Center for Information
     8	and Neural Networks (CiNet), National Institute of Information and Communications Technology (NICT), Suita, Osaka 565-0871, Japan; 3 Advanced
     9	Telecommunications Research Computational Neuroscience Laboratories, 2-2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan; 4 Nagahama Institute of
    10	Bio-Science and Technology, 1266 Tamura-cho, Nagahama, Shiga 526-0829, Japan
    11	†
    12	Naotsugu Tsuchiya, http://orcid.org/0000-0003-4216-8701
    13	‡
    14	Hayato Saigo, http://orcid.org/0000-0002-4209-352X
    15	*Correspondence address. Biomedical Imaging facility, Clayton, School of Psychological Sciences and Turner Institute for Brain and Mental Health, Monash
    16	University, 770 Blackburn Monash, VIC 3168, Australia. Tel:+61-3-9905-4564; E-mail: naotsugu.tsuchiya@monash.edu.
       
    17	Abstract
    18	Characterizing consciousness in and of itself is notoriously difficult. Here, we propose an alternative approach to characterize, and eventually define, consciousness through exhaustive descriptions of consciousness’ relationships to all other consciousness. This approach
    19	is founded in category theory. Indeed, category theory can prove that two objects A and B in a category can be equivalent if and only
    20	if all the relationships that A holds with others in the category are the same as those of B; this proof is called the Yoneda lemma. To
    21	introduce the Yoneda lemma, we gradually introduce key concepts of category theory to consciousness researchers. Along the way, we
    22	propose several possible definitions of categories of consciousness, both in terms of level and contents, through the usage of simple
    23	examples. We propose to use the categorical structure of consciousness as a gold standard to formalize empirical research (e.g. color
    24	qualia structure at fovea and periphery) and, especially, the empirical testing of theories of consciousness.
       
    25	Keywords: consciousness; contents of consciousness; level of consciousness; qualia; category theory; yoneda lemma; functor; natural
    26	transformation
       
    27	Problems in characterizing consciousness
    28	on its own
    29	Over the last few decades, the question of the nature of consciousness is gaining a respected position as the target of scientific
    30	inquiry. Much of the empirical research has tried to identify the
    31	neural correlates of consciousness (Koch et al. 2016). Building on
    32	the massive amount of empirical evidence, some models or theories have been proposed to explain the link between various
    33	features and aspects of the neural activity and the associated
    34	functions and phenomena of consciousness (Engel and Singer
    35	2001; Graziano and Kastner 2011; Lamme 2015; Northoff and
    36	Huang 2017; Brown et al. 2019; Mashour et al. 2020).
    37	Another distinct approach to consciousness is to study it using
    38	a mathematical framework (Hoffman 1966, 1980; Stanley 1999;
    39	Tononi 2004; Fekete and Edelman 2011; Yoshimi 2011; Prentner 2019; Kleiner 2020; Prakash et al. 2020; Signorelli et al.
    40	2021). Among these, the Integrated Information Theory (IIT) of
       
    41	consciousness by Tononi and colleagues (Oizumi et al. 2014;
    42	Tononi et al. 2016) is arguably the most developed and discussed
    43	in the literature. IIT takes a unique approach, where it tries
    44	to first identify the essential properties of consciousness, which
    45	are always true to any experience. These essential properties
    46	are called “phenomenological axioms,” and IIT tries to derive
    47	mathematical postulates that any physical system should satisfy to support these properties. IIT then proposes an explanatory
    48	“identity” between phenomenal consciousness and information
    49	structure (Haun and Tononi 2019).
    50	In the past (Tsuchiya et al. 2016), we, the authors, have suggested that, rather than trying to propose the explanatory “identity” between the two in a single step, it would be empirically
    51	more tractable to break up the IIT project into the following
    52	three subprojects: (i) to characterize the structure of conscious
    53	phenomenology as a category; (ii) to characterize the structure
    54	of information as a category; and (iii) to assess the degree of
    55	similarity between the structures. For the second issue, we and
       
    56	Received: 9 March 2021; Revised: 18 August 2021; Accepted: 24 September 2021
    57	© The Author(s) 2021. Published by Oxford University Press.
    58	This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which
    59	permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
       
    60	Downloaded from https://academic.oup.com/nc/article/2021/2/niab034/6397521 by guest on 14 January 2026
       
    61	A relational approach to consciousness: categories of
    62	level and contents of consciousness
       
    63	2
       
    64	Tsuchiya and Saigo
    65	(including B) are the same (up to natural equivalence defined in
    66	“Functor category, whose objects and arrows are functors and natural transformations”) as those between B and the rest (including
    67	A). Importantly, this is true, even if A and B themselves are difficult to characterize in and of themselves, as in the case of black
    68	holes in cosmology or infinity in mathematics. While defining consciousness directly in a way where everyone agrees is not easy,
    69	characterizing consciousness through a rich set of relationships
    70	is much more feasible.
    71	Note that we will not offer a definitive definition(s) of consciousness in this article. Rather, our plan is to introduce a novel
    72	perspective on how we can start doing so through the application of the Yoneda lemma. Through our proposed approach,
    73	eventually, we would hope to completely justify the relational
    74	definition of consciousness. But first things first. To apply the
    75	Yoneda lemma, we need to propose several possible categories of
    76	consciousness. After introducing key concepts in category theory,
    77	we will come back to the issue of how we can apply the Yoneda
    78	lemma in consciousness research and discuss what it means for
    79	consciousness research. For example, the Yoneda lemma will
    80	enable us to address the question of the equivalence of color
    81	qualia structure at the fovea and the periphery in a systematic
    82	way.1
       
    83	What’s the category in category theory?
    84	Categories of consciousness
    85	In this section, we will explain the basics of category theory,
    86	which is necessary to explain the Yoneda lemma with a view to
    87	its application in consciousness research. We will propose two
    88	categories: category of level of consciousness, Lv, and category
    89	of contents of consciousness, Q Note that we are not claiming
    90	that these examples are “the” only possible formulation of categories of consciousness. They are invoked here as a starting point.
    91	For an accessible introduction to category theory, see Lawvere
    92	and Schanuel (2009); Spivak (2014); Bradley (2017); Northoff et al.
    93	(2019).
    94	The basis of category theory is three concepts: (i) category,
    95	(ii) functor, and (iii) natural transformation. Let us start with a
    96	category.
    97	Definition: For a collection of objects to be considered as a category,
    98	they must satisfy the following five axioms.
    99	1. An arrow has its “source” object called domain and “target” object
   100	called codomain.
   101	2. For every object, there is a self-referential arrow called identity.
   102	3. A pair of arrows is composable if the domain of one arrow equals
   103	the codomain of another.
   104	4. Identities do not change other arrows by composition.
   105	5. Composition is associative.
   106	In other words, a category is a system consisting of “objects”
   107	and “arrows.” Figure 1 explains the above definition with a diagram
   108	in an intuitive manner.
   109	For a category of level of consciousness, we consider global
   110	states of consciousness or the degree of wakefulness as an object.
   111	The level of consciousness is usually assumed to go from 0 in dead
   112	humans, which is lower than deeply anesthetized or dreamless
   113	1
   114	We assume that many readers of this article are unfamiliar with category theory and have no or little background training in mathematics. These
   115	readers are our primary target audience. To make the mathematical aspects as
   116	accessible as possible, we provided informal interpretations of mathematical
   117	concepts and their concrete examples in the context of consciousness research.
   118	We found this highly effective in introducing category theory to scientists with
   119	no or little mathematical background (Spivak 2014).
       
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   121	others have made some initial efforts on the mathematical side
   122	(Northoff et al. 2019; Kleiner and Tull 2020; Tull and Kleiner 2020).
   123	Notably, there has been progress on the application of category
   124	theoretical approaches toward stochastic processes (Manin and
   125	Marcolli 2020) to formalize information structures (Baudot and
   126	Bennequin 2015; Baudot et al. 2019). For the third issue, we also
   127	made an initial attempt (Haun et al. 2017). Here, we deal with the
   128	first issue: how we can characterize the structure of consciousness as a category. Note that category theory has been proposed to
   129	model aspects of cognition and how they are supported by neural
   130	networks in the brain (Ehresmann and Vanbremeersch 1987, 1996;
   131	Arzi-Gonczarowski 1999; Healy et al. 2008; Phillips and Wilson
   132	2010, 2016; Ehresmann and Gomez-Ramirez 2015; Phillips 2020).
   133	Precisely defining consciousness has been notoriously difficult (Dennett 1988; Sloman 1991; Stanovich 1991; Chalmers 1996;
   134	Velmans 2009; Kleiner 2020), but certain characterizations of consciousness are accepted by many for use in empirical consciousness research (Koch et al. 2016; Mashour et al. 2020). One of the
   135	widespread consensus is to distinguish the “level” and “contents”
   136	of consciousness (Laureys 2005; Boly et al. 2013). For example,
   137	Searle (Searle 2000) gives the following definition: “Consciousness
   138	consists of inner, qualitative, subjective states and processes of
   139	sentience or awareness.” Consciousness, so defined, begins when
   140	we wake in the morning from a dreamless sleep and continues
   141	until we fall asleep again, die, go into a coma, or otherwise become
   142	“unconscious.” In the case of the contents of consciousness, a particular conscious experience of red color, often called a red quale
   143	(Kanai and Tsuchiya 2012), is typically characterized as “redness
   144	of red” or “red like this wine.” In these cases, consciousness is
   145	characterized in reference to another conscious experience. This
   146	relational nature of consciousness appears to be one of the fundamental characteristics of consciousness (Nagel 1974; Chalmers
   147	1996; Kleiner 2020).
   148	In this paper, by introducing several notions of category theory,
   149	we offer mathematical justification for the relational characterization of consciousness. In fact, situations where a particular
   150	object is not possible to define or even characterize on its own
   151	arise relatively often outside of consciousness research. In these
   152	situations, what has been effective is to rely on the relationships between the object to be defined and its surroundings. For
   153	example, some linguists consider that meanings of a word can be
   154	understood only through how the word is related to other words
   155	and how they are put into the context in the sentence (e.g. Frege
   156	1980; De Saussure 2011). In ecology, it is essential to characterize
   157	any life form within an ecosystem; defining a tree without mentioning the geological area in which it lives and which animals,
   158	insects, and other plants interact with it and in what way would
   159	miss the very essence of what that tree is. In quantum theory,
   160	the essence of quantum features is that they are only possible
   161	to explain through the interactions between the objects (Coecke
   162	and Kissinger 2017). In cosmology, black holes are, in themselves,
   163	impossible to measure and characterize, but their interaction with
   164	their neighbors can be measured and used to characterize their
   165	properties. In mathematics, various types of infinity can be distinguished through what types of relationships they have with other
   166	mathematical objects.
   167	There is a deep mathematical foundation in why these theoretical fields can dispense with a direct definition of an object in
   168	favor of characterizing its interaction with its surroundings. That
   169	is what we introduce to consciousness research in this paper: the
   170	Yoneda lemma. In short, the Yoneda lemma allows us to equate
   171	A with B (up to isomorphism defined in “What does it mean for
   172	objects to be the same in a category? Isomorphic objects in a
   173	category”) if the relationships between A and the rest of objects
       
   174	A relational approach to consciousness
       
   175	Figure 1. Requirements of a category. (a) Composition: if A, B, and C are
   176	objects in category X, and f: A → B and g: B → C are arrows in category
   177	X, then we can compose (or combine) f and g to obtain an arrow, f;g: A →
   178	C. Note f;g reads as “f then g” and it is often denoted as g◦ f (Fong and
   179	Spivak 2019). (b) Associativity: if f, g, and h are arrows in category X, then
   180	the order to compose the arrows does not matter: (f;g);h = f;(g;h). A, B, C,
   181	and D are objects in category X. (c) Unit: For any object A in category X,
   182	there is a self-referential arrow A → A, which is called identity arrow: 1A.
   183	For any arrow f: A → B, the following is always satisfied: 1A;f = f = f;1B
       
   184	A category of level of consciousness
   185	Let us verify whether our proposed objects and arrows for level of
   186	consciousness, Lv, can qualify as a category. Objects are certain
   187	levels of consciousness supported by a human brain, such as A, B,
   188	C, etc. An arrow f exists between two conscious levels A and B, if
   189	level of consciousness A “is higher than or equal to” B.
   190	Consider the requirement of composition. Suppose that f: A
   191	→ B and g: B → C exist as arrows for objects A, B, and C in consciousness level category Lv. That is, the level of consciousness
   192	A is higher or the same as B, and B is higher or the same as C.
   193	Then, it follows that A is higher than C or the same. In other words,
   194	there is an arrow from A to C, which is the combination of f and g,
   195	which we denote as f;g. Therefore, the condition for composition
   196	is satisfied.
   197	Next, let us consider associativity. We assume that for levels
   198	of consciousness A, B, C, and D, three arrows exist f: A → B, g: B
   199	→ C, and h: C → D. Then, (f;g);h means that (i) we first confirm
   200	that A is higher than or the same as C (i.e. f;g) and (ii) due to h
       
   201	(i.e. C is higher than or equal to D), we conclude that A is higher
   202	than or the same as D. On the other hand, f;(g;h) means that (i) we
   203	first confirm g;h, that is, B is higher than or equal to D, and (ii) we
   204	conclude A is higher than or equal to D. In other words, associativity requires these two conclusions to be exactly the same. In the
   205	diagram (Fig. 1b), this translates to the fact that two conclusions
   206	via two paths are exactly the same if the starting and the ending
   207	points are the same.
   208	Finally, let us check the unit. This is the reason why we have
   209	added “or equal to” as a part of the definition of the arrow. For any
   210	level of consciousness A, there exists an arrow 1A: A → A because
   211	A is always higher than or equal to A. If f:A → B exists, then If f:
   212	A → B exists, then 1A;f means the following. (i) We confirm that
   213	A is the same as A. Then (ii) we conclude that A is higher than or
   214	equal to B. Similarly, f;1B means the following. (i) We confirm A
   215	is higher than or equal to B. Then, (ii) we confirm B is equal to B.
   216	If we define f: A → B as A is strictly higher than B, our proposed
   217	system of objects and arrows for level of consciousness Lv does
   218	not qualify as a category.
   219	Our category of level of consciousness Lv is an example of categories called “preordered sets.” Preordered sets are categories such
   220	that between any two objects there is at most one arrow. The
   221	arrows in preordered sets are called “preorder.”
       
   222	A category of contents
   223	Next, let us examine if our proposed definition can make a system
   224	of contents of consciousness Q into a category. Here, objects are
   225	the contents of consciousness. Let us consider a system Q that
   226	consists of only three objects, A = red sunset, B = red crayon, and
   227	C = red wine. For simplicity, we consider a case of a preordered set;
   228	that is, between two objects, there is either one arrow or none at
   229	all. Suppose there is an arrow from A to B when the contents of the
   230	two consciousness are “nearly indistinguishable” in terms of their
   231	color [for those who are concerned about the gradual degradation
   232	of distinguishability, see Tsuchiya et al. (2021)]. In other words, if
   233	there is an arrow f between the two contents of consciousness A
   234	and B, then f: A → B, and A is subjectively felt as nearly the same
   235	as B in terms of their color. It is clear that the composition holds.
   236	In other words, if the redness of sunset and crayon are similar,
   237	and the redness of crayon and wine are similar, then the redness
   238	of sunset and wine are also similar. Mathematically speaking, if
   239	f: A → B and g: B → C, then f;g: A → C also holds. Associativity
   240	holds as well. The unit is also valid (these two proofs are simple
   241	and left to the reader). Therefore, our proposed system of objects
   242	and arrows Q can constitute a category.
   243	Here, since arrows mean “nearly indistinguishable,” the direction of an arrow does not matter. Now that we are considering a
   244	preorder, there is only one arrow in either direction. This implies
   245	that any arrow in this category is invertible. That is, any arrow has
   246	an arrow of a reverse direction such that the composition of the
   247	two is the identity arrow. Invertible arrows are also called “isomorphism” not only in preordered sets but also in categories in
   248	general. When all arrows are isomorphism, the category is called
   249	“groupoid.” In sum, we proposed an exemplar category Q, which is
   250	preorder and groupoid, where all arrows are isomorphism (if there
   251	are multiple isomorphisms between objects, then such a category
   252	is groupoid but not preorder).
       
   253	What does it mean for objects to be the same in a
   254	category? Isomorphic objects in a category
   255	So far, we have introduced categories Lv and Q for level and contents of consciousness. Next, to deepen our understanding of
       
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   257	sleep, and highest in fully wakeful states (Mormann and Koch
   258	2007). Note that one object can contain many elements. As an
   259	example, let us consider a fully wakeful state as an object. One
   260	can distinguish each moment of experience as different elements.
   261	An object in this case is a group of these elements considered as
   262	one entity, much like the colloquial use of a “category.” Mathematically speaking, an object does not have to be one element.
   263	For a category of contents, we can consider any possible content of consciousness, such as color, sound, and pain. As is usually
   264	the case in mathematics, however, starting from the most general
   265	situation, encompassing all levels of consciousness in all animals
   266	or all types of contents, is not a wise strategy to make progress.
   267	Accordingly, we will consider a very concrete case of consciousness categories below and leave generalization of these categories
   268	to future work.
   269	An arrow, →, relates an object with another object. For example, f: A → B denotes a relationship f between object A and object
   270	B. In the case of level of consciousness category Lv, we can define
   271	the meaning of f to be the level of consciousness in A “is higher or
   272	equal to” B. Soon, it will become clear why we need “equal to.” Read
   273	on! In the case of content category Q, we can define the meaning
   274	of f to be that A “is experienced as nearly indistinguishable with”
   275	B in a certain aspect.
   276	If the system consisting of objects and arrows satisfy further
   277	three conditions (composition, associativity, and unit) as depicted
   278	in Fig. 1, then it qualifies as a category. We visualize these three
   279	requirements concisely and intuitively using diagrams. This is one
   280	of the most powerful aspects of category theory: simply visualizing complex relationships to facilitate the understanding of the
   281	subject matter.
       
   282	3
       
   283	4
       
   284	Tsuchiya and Saigo
   285	conscious states, where we can never subjectively compare which
   286	state is “higher” as conscious level. For example, which is higher
   287	in level of consciousness between vivid dreaming and drowsy
   288	awakening states? Or, what about between deep general anesthesia and coma, where consciousness completely disappears?
   289	We believe that these issues arise because the scholars who criticize the concept of level of consciousness implicitly assume that
   290	“level” is something that is “isomorphic” to natural numbers (or
   291	positive real numbers), where it is always possible to rank the
   292	order between two objects. Such a strongly ordered structure is
   293	called “total order” in mathematics. We believe that the total
   294	order assumption is not necessary at all for the concept of level
   295	of consciousness to be useful. We propose that a looser concept,
   296	preorder, is a useful and appropriate concept for level of consciousness. As for preorder, any two objects (levels) may or may
   297	not have a relationship “≦.”
   298	Our proposed category Lv can be applied to many cases. Compared to the fully wakeful states, deeply anesthetized, dreamless
   299	sleep, or coma would be lower in the level of consciousness. In Lv,
   300	we will have no arrows between cases of lower level of consciousness, where subjects have no ability to compare their levels and
   301	do not show any behavioral outputs to be meaningfully compared.
   302	Likewise, Lv can be proposed across animals without proposing
   303	any arrows between animal species but only considering arrows
   304	within animals (between fully wakeful and deeply anesthetized,
   305	deep sleep, or coma).
   306	Note that the two categories Lv and Q proposed here are just toy
   307	examples and we are not claiming that they are “the” categories
   308	of consciousness. In both level and contents of consciousness, it
   309	is possible to focus on other aspects of consciousness and consider the corresponding categories. For example, by focusing on
   310	an aspect of consciousness that changes over time, one can propose a “category of mobility” (Saigo et al. 2019). Another kind
   311	of category, called “co-slice category,” is effective in capturing
   312	association possibilities and metaphorical structures of meaning (Fuyama and Saigo 2018) [note that the subjective experience
   313	of “meaning” of a word is not accepted widely, however; see
   314	Kemmerer (2015) and McClelland and Bayne (2016) for recent discussions on this issue]. Tsuchiya et al. (2021) explains a further
   315	framework to loosen the condition of arrows in Q by making it
   316	in variable degrees of similarity using a concept of enriched categories (Lawvere 1973; Leinster and Meckes 2017; Fong and Spivak
   317	2019).
   318	The point of this section was to provide the readers an understanding that it is not difficult to propose a category of consciousness. Some readers may think that as a mathematical theory,
   319	category theory cannot be applied to the problem of consciousness, especially because of the problem in composition (Pitt 2018).
   320	As we explained, however, category theory can be applied if
   321	“objects” and “arrows” are appropriately defined and if they satisfy
   322	a few conditions.
       
   323	Preliminary conclusion and discussion
       
   324	In simple terms, a functor is an arrow between two categories (for
   325	a graphical definition, see Fig. 2). Importantly, a functor needs to
   326	map one category to another category while keeping the “structure” of the category (e.g. commutativity). From the viewpoint
   327	of equivalence discussed in the last chapter, “the existence of a
   328	functor” is one important condition to consider as the sameness
   329	between categories. The sameness as “existence of a functor” is
   330	much weaker than the other types of sameness discussed in this
   331	paper.
       
   332	We believe that introducing the concept of preordered sets
   333	as a category of level of consciousness Lv has an important implication in consciousness research. Some authors
   334	have raised issues with the concept of level of consciousness (Bayne et al. 2016; Pautz 2019). One of the issues is
   335	that this concept seems to imply a gradual change from
   336	unconsciousness to full consciousness, which is not widely
   337	agreed upon. Another issue is the existence of certain pairs of
       
   338	What is a functor?
   339	A functor is an arrow between categories
       
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   341	what it means to define an object, we will consider what it means
   342	for objects A and B to be equivalent.
   343	When we say that A and B are the “same” in everyday life,
   344	it means that they are equivalent with respect to some definitions or assumptions and ignoring various aspects. For example,
   345	when there are two apples on the right and the left, they may
   346	be similar in terms of their color or a category within the fruit,
   347	but they are different in their surface textures. In fact, they are
   348	not the same thing, being located in different places. Category
   349	theory provides a wealth of mathematical tools to handle these
   350	subtle differences in “sameness” (Tsuchiya et al. 2016) in that it
   351	characterizes different kinds of sameness using various types of
   352	relationships.
   353	Let us explain the sameness between objects in a category,
   354	called “isomorphism.”
   355	Definition: The objects A and B in a certain category C are isomorphic if there is an “invertible” arrow (called isomorphism) between them.
   356	When the arrow f: A → B is “invertible,” there exists an arrow g: B → A
   357	with f ;g = 1A and g;f = 1B.
   358	In the case of category Lv, if coma level A is lower than minimally consciousness level B, the reverse of the arrow f: A → B
   359	does not exist. In our proposed example of the category of contents with three objects, all arrows are isomorphisms (see Section
   360	2.3), that is, any pair of objects is isomorphic if there exists an
   361	arrow between them.
   362	This concept of isomorphism is much looser and more flexible than the usual concept of “equalities.” For example, under
   363	the framework of set theory, a set of alphabets {a, b, c, d, e} is
   364	not equal to a set of numbers {1, 2, 3, 4, 5}. The difference in
   365	terms of “elements” disqualifies them to be considered as equal
   366	sets. Now, consider a category Set whose objects are sets and
   367	arrows are functions or mappings between sets. Within the category Set, two objects (i.e. sets) {a, b, c, d, e} and {1, 2, 3, 4, 5}
   368	are considered as isomorphic objects in the sense that each element can be mapped from one set to the other set in a one-to-one
   369	manner. In fact, a concept of a number, such as 5, is “defined”
   370	when we regard these “different” sets as the “equivalent” in a certain respect. In this sense, isomorphism captures the concept of
   371	“essential sameness” in the category. As another example of isomorphism between objects in a category, consider a category of
   372	topological space, Top, which is the foundation of the mathematical field called topology. In Top, objects are topological spaces
   373	and arrows are continuous maps. While coffee cups and donuts
   374	are usually considered as “completely different” objects, they are
   375	isomorphic in Top.
   376	In order to make progress in consciousness research, equivalence in the sense of isomorphism is not powerful enough. As
   377	we discuss later, we believe that different types of sameness in
   378	category theory will be more useful and find wider application in
   379	consciousness research. In particular, isomorphism between categories, or even looser sameness, called categorical equivalence
   380	will be critical. To introduce these concepts, we need to introduce
   381	two concepts: functors and natural transformations, which are
   382	the topics of the next two chapters.
       
   383	A relational approach to consciousness
       
   384	Definition: A mapping F, which maps an object and an arrow in category C to an object and an arrow in category D, is called a functor if it
   385	satisfies the following three conditions:
   386	1) F(1X) = 1 F(X), that is, the identity arrow of X will map to the
   387	identity arrow of F(X).
   388	2) f:X → Y will map to F(f):F(X) → F(Y), that is, the arrow from X to
   389	Y maps to the arrow from F(X) to F(Y).
   390	3) a composite arrow f;g in category C is preserved as F(f;g) = F(f);F(g).
   391	To see the relevance of functors in consciousness research, let
   392	us examine a functor from categories of level of consciousness Lv.
   393	To remind you, Lv is a preordered set; objects and arrows in Lv are
   394	level of consciousness and “≦” as introduced previously. As we
   395	argued ‘there’ the construction of category Lv and clarifying its
   396	specific structure itself is a difficult research program in itself. But
   397	here, we focus on the other problem: to establish a consciousness
   398	meter, which translates to a construction of a functor from category Lv to a convenient preorder category D. As an example,
   399	consider a category D whose objects and arrows are natural numbers and ≦. Readers can check easily that D is a preorder category;
   400	any pair of objects have at most one arrow (e.g. 1 ≦ 2), any arrows
   401	can be composed in an associative manner (e.g. 1 ≦ 2 ≦ 3) and
   402	that each object has a unit arrow (e.g. 1 ≦ 1).
   403	Let us consider a mapping F that maps from category Lv to category D and satisfies the conditions to be a functor. In other words,
   404	F maps any object of Lv, that is, any level of consciousness, into a
   405	natural number, while all ≦ relations in Lv are preserved in D. In
   406	other words, searching for a functor F is the same as the construction of a consciousness meter as strived for by some consciousness
   407	researchers. We also note here that mapping in the other direction, that is, from D to Lv does not have to be a functor. In other
   408	words, the validity of a consciousness meter, F, is not compromised if two objects with one arrow (e.g. 1 ≦ 2 in D) do not map
   409	onto Lv functorially. For example, from D to Lv, a mapping G may
   410	map 1 to coma and 2 to deep general anesthesia. If G does not map
   411	≦ in D into ≦ in Lv, then G is not a functor. However, as we pointed
   412	out in the last section, it is unclear whether ≦ exists between
   413	coma and deep anesthesia; thus, this itself is not a problem for
   414	F as a valid consciousness meter.
       
   415	Figure 3. Exemplar functors between categories QX and QY. (a) In
   416	category QX, there exist arrows for all possible object pairs (only three
   417	arrows are depicted here). In category QY, there are no arrows except for
   418	identity arrows for each object (not drawn). (b) One of 27 (=33 ) functors
   419	from QY to QX that preserve structures of QY. (c) One of three functors
   420	from QX to QY. This functor has to map all objects in QX into one object
   421	in QY and all arrows in QX to the identity arrow in QY
       
   422	Next, let us consider a functor F for category Q for contents of
   423	consciousness as considered in the last section. Q is a groupoid,
   424	and its three objects are A = red sunset, B = red crayons, and
   425	C = red wine. Arrows are “nearly indistinguishable” in terms of
   426	color. Here, we consider two persons’ categories QX and QY. For
   427	Ms X, all objects’ colors are nearly indistinguishable; thus, there
   428	are arrows for any pair of objects AX, BX, and CX. However, for Mr
   429	Y, all objects are distinguishable; thus, there are no arrows among
   430	AY, BY, and CY (Fig. 3a).
   431	In this situation, there are 27 functors that map from category
   432	QY to category QX since the destination of each object in category
   433	QY can be any of AX, BX, and CX. For example, one functor F1
   434	maps AY to AX, while it maps BY and CY into BX (Fig. 3b). The only
   435	arrows in QY are the identity arrows. The identity for AY maps
   436	to that for AX, while those for BY and CY maps to that for BX.
   437	The requirement of composition is also satisfied (because the only
   438	way to compose arrows in QX is to compose the identity with the
   439	identity, which is the identity).
   440	What about functors from category QX to category QY? Like
   441	the above example, if F collapses all objects in QX into one
   442	object in QY, F can satisfy all conditions to be a functor. If F
   443	does not collapse them, what happens? For example, if AX is
   444	mapped to AY, while BX is mapped to BY. Because Mr Y can
   445	distinguish AY and BY, there is no arrow between AY and BY.
   446	However, Ms X sees AX as indistinguishable from BX. Thus, the
   447	arrow that connects AX and BX is lost in translation during this
   448	mapping. Such a faulty mapping does not qualify as a functor.
   449	From a viewpoint of equivalence, there is a certain “structural
   450	similarity” in subjective similarity characterized by categories QX
   451	and QY in a sense that there exists a functor from one category to the other. But the richness of the structure is “different” as can be quantified from the number of arrows in each
   452	category. This nicely represents the situation of the subjective
   453	experience of Ms X and Mr Y—they experience the objects with
   454	certain structural similarity and difference in richness in the
   455	structure.
   456	In sum, functors can be used to compare structures between
   457	categories. In terms of categories of contents of consciousness Q,
   458	functors are not limited in the usage of comparison of similarity
   459	structures among individuals as in our examples. Functors can be
       
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   461	Figure 2. What is a functor? A structural mapping F between category C
   462	to D is called a functor if the following three conditions are satisfied:
   463	(i) an arrow f: X → Y in category C is mapped onto a corresponding
   464	arrow F(f): F(X) → F(Y) in category D; and (ii) a composite arrow f;g in
   465	category C is preserved as F(f;g) = F(f);F(g); and (iii) an identify arrow 1X
   466	for an object X in category C is preserved as F(1X) = 1 F(X)
       
   467	5
       
   468	6
       
   469	Tsuchiya and Saigo
       
   470	used to compare structures of content category of color with content category of shape, content category of meaning, etc. within
   471	individuals. This is one of the possible future directions of category
   472	theoretical research of conscious contents.
       
   473	Category of categories
       
   474	What is a natural transformation?
   475	A natural transformation is the central concept in category theory.
   476	Indeed, category theory was originally invented to introduce the
   477	concept of natural transformation. To explain natural transformation, the concept of functor was introduced. And to explain the
   478	concept of functor, the concept of category was introduced. Natural transformations are necessary to state and prove the Yoneda
   479	lemma. The Yoneda lemma and its inspiration for consciousness
   480	research is the central message of the paper: even if it is difficult
   481	or impossible to characterize consciousness per se, we can do so
   482	by characterizing all arrows in the relevant category.
       
   483	A natural transformation is an arrow between
   484	functors
   485	A natural transformation maps a functor to another functor while
   486	keeping the structure of the functor. The basic structure of category theory can be stated in terms of categories, functors, and
   487	natural transformations. While there are more, solid understanding up to natural transformation is critical to handle category
   488	theory. It is important to develop an intuition about a natural
   489	transformation. It is a family of arrows in the category, which a
   490	functor maps the original category into. The graphical definition
   491	in Fig. 4 helps understand this concept.
   492	Definition: Consider functors F and G that map category C to category D. When a mapping t from functor F to G satisfies the following two
   493	conditions, t is called a natural transformation and we write t: F G.
   494	1. Given an object X in category C, t gives an arrow tX: F(X) → G(X)
   495	in category D.
   496	2. For any arrow f: X → Y in category C, tX;G(f) = F(f);tY holds in
   497	category D.
   498	tX is called the X component of t.
       
   499	Natural transformation for category Lv
   500	To understand the relevance of natural transformations in
   501	consciousness research, let us first consider the category of level
   502	of consciousness Lv and the functors. Suppose Ms F proposed a
   503	functor F as a consciousness meter, which maps all levels in preorder category Lv to preorder category D (objects and arrows are
       
   504	Figure 4. What is a natural transformation? (a) Category C (consisting of
   505	objects X and Y and an arrow f) is mapped into category D by two
   506	functors F and G. Each functor preserves the structure of category C in D
   507	(e.g. F(f):F(X) → F(Y), G(f):G(X) → G(Y)). Natural transformation t maps
   508	functor F into G while preserving its structure. t can be considered as a
   509	family of arrows in category D, such as tX and tY. Each arrow is specified
   510	by an object in the original category C, such as tX, which maps an object
   511	F(X) to G(X) in category D. b) Visualizing t as a family of arrows in the
   512	destination category D
       
   513	natural numbers and ≦). F maps all objects in Lv to numbers in
   514	D and ≦ in Lv into ≦ in D. On F’s monitor, level of consciousness
   515	under deep general anesthesia shows up as number 10, while that
   516	under the wakeful state is 100. Then Mr G also developed another
   517	functor G as a consciousness meter. On G’s meter, the level under
   518	deep general anesthesia is 20 and the wakeful state is 200. In both
   519	cases, the arrow in category C is preserved as the arrow in category D as ≦. Now, we ask: what is the nature of the relationship
   520	between F’s and G’s consciousness meters?
   521	Let us assume that there exists a natural transformation t
   522	between functors F and G. Then, according to Condition 1 above,
   523	the natural transformation t gives us two arrows tA and tW for two
   524	objects that we considered in category Lv: A for anesthesia and W
   525	for wakefulness. tA and tW are two arrows in category D, which
   526	satisfies the following: tA: F(A) = 10 → G(A) = 20, tW: F(W) = 100 →
   527	G(W) = 200. Interestingly, this arrow corresponds to ≦, the arrow
   528	in D.
   529	Moreover, according to Condition 2, with respect to two objects
   530	A and W and the arrow f:A → W in category Lv, we have
   531	tA;G(≦) = F(≦);tW. The left-hand side of this equation means that
   532	(i) we first evaluate the value for anesthesia in F (=10) to translate
   533	into the value in G (=20), then (ii) we confirm the relation between
   534	A and W (≦) in category Lv to be translated into the relation in G’s
   535	monitor (20 ≦ 200). The right hand side of the equation means
   536	that (i) we first translate the relation A ≦ W in category Lv into
   537	category D (F(A) = 10 ≦ F(W) = 100) and then (ii) we map the value
   538	for wakefulness in F’s monitor (=100) into G’s monitor (=200). In
   539	sum, this family of arrows in category D guarantees a lawful relationship between two functors, preserving the original relations
   540	in category Lv (note that we also have two more relations with
   541	respect to the identity arrows for 1A and 1 W in category Lv).
       
   542	Natural transformation for category Q
   543	Next, let us consider a natural transformation for the category of
   544	contents Q. We consider two categories based on Q introduced in
   545	the section on “A category of contents” and consider a situation
   546	that can be tested in a psychophysical experiment. For simplicity,
   547	we consider two objects as subjective experience of a red Apple (A)
   548	and a red Berry (B) and arrows as a relationship for “nearly indistinguishable” in terms of color (Fig. 5). Category C refers to objects
       
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   550	In a category of categories, Cat, objects and arrows are categories
   551	and functors (and this is the origin of the title for the last section).
   552	This is an interesting characteristic of category theory; depending
   553	on the viewpoint, what we are talking about can be considered as
   554	objects in a certain category or arrows in another category. Furthermore, a functor, which is considered as an arrow between two
   555	categories in this section, will be considered as an object in the
   556	later section. All of this comes from the fact that category theory
   557	is mathematics that focuses on “relationships.”
   558	Let us consider a category of categories using an example of
   559	category of content Q. We ask Ms X and Mr Y to come back on the
   560	stage. If we consider category QX and QY as objects and functor
   561	F as arrows, the example in Fig. 3 is an exemplar category of
   562	category Q.
       
   563	A relational approach to consciousness
       
   564	7
       
   565	Functor category, whose objects and arrows are
   566	functors and natural transformations
       
   567	and arrows in the Central visual field, while category E refers to
   568	those in the Entire visual field, which includes the central, left,
   569	and right visual field. A mapping R from C to E translates objects
   570	and arrows to those in the Right visual field. Likewise, a mapping L
   571	from C to E translates those in the Left visual field. The mappings L
   572	and R are likely to be functors in healthy subjects. One way to test
   573	this idea is to operationally define category C with objects as a set
   574	of perceptual objects and arrows as similarity relationships. Likewise category E, which includes C but also across the entire visual
   575	field. Under this condition, psychophysical experiments measuring similarity relationships can test if the mapping L and R satisfies
   576	the functor conditions. In the following, we assume L and R satisfy
   577	the functor conditions.
   578	Natural transformation t can clarify the relationship between
   579	functors L and R. Let us go through this example slowly as
   580	this is an important point. Assume the existence of a natural
   581	transformation t from functor L to functor R. Now, with respect
   582	to objects A (Apple) and B (Berry) as well as its arrow f (f:A → B; A
   583	is nearly indistinguishable from B in terms of color) in category
   584	C for Central visual field, a natural transformation is a collection of arrows in category E for Entire visual field. These arrows
   585	include tA: L(A) → R(A) and tB: L(B) → R(B) (Condition 1), both of
   586	which mean “nearly indistinguishable” in terms of color, which
   587	are indeed the arrows in category E. tA means that Apple is nearly
   588	indistinguishable in terms of color in the Left and Right visual
   589	field.
   590	Condition 2 says for an arrow f:A → B in category C, we have
   591	tA;R(f) = L(f);tB. The left side of this equation first confirms that
   592	Apple is indistinguishable between Left and Right visual fields,
   593	then it confirms that Apple in Right is indistinguishable from Berry
   594	in Right. The right side of the equation first confirms that Apple is
   595	indistinguishable from Berry in Left, then Berry in Left is indistinguishable from Berry in Right. In other words, Condition 2 means
   596	that the color of Left Apple is indistinguishable from the color of
   597	Right Berry, and it allows us to go in either way to prove this fact.
   598	The power of the natural transformation t is to handle a lot of
   599	arrows, such as tA and tB, all at once in a lawful manner.
       
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   601	Figure 5. Natural transformation. Objects are Apples (A) and Berries (B)
   602	in categories C (central visual field) and E (entire visual field). Arrows in
   603	categories C and E are “nearly indistinguishable” in terms of color. Thus,
   604	category C is included in E. Central vision (C) is mapped to Left and Right
   605	visual fields by functor L and R, respectively, preserving objects and
   606	arrows. If natural transformation t exists from functor L to functor R,
   607	then indistinguishability of Left Apple to Right Berry can be established
   608	by both through Left Berry (=L(f);tB) or Right Apple (=tA;R(f)), that is,
   609	there is no path dependency. Note that tA is an arrow in category E,
   610	indicating that Apple in the Left is “nearly indistinguishable” from Apple
   611	in the Right. Same goes with tB for Berry in the Left and Right
       
   612	Using the concept of natural transformation, we introduce
   613	another critical concept to prove the Yoneda lemma: functor category. Functor categories and equivalence defined in this manner,
   614	we suspect, is something that is likely to be completely missing in
   615	current language in consciousness research, and it is very useful
   616	for future conceptual analyses.
   617	Definition: Functor category, Fun(C, D), considers objects and arrows
   618	as functors from category C to category D and natural transformations
   619	between them.
   620	From the viewpoint of the contents category, let us consider
   621	a functor category. We extend the example categories C and
   622	E in Fig. 5. Simply, treating functor L and R as objects and a
   623	natural transformation t as an arrow constructs a functor category Fun(C, E).
   624	What is the benefit of considering a functor category? One of
   625	the benefits is to allow us to introduce two concepts: “natural
   626	equivalence” and “categorical equivalence.” Categorical equivalence is likely to be a critical target to establish in empirical studies
   627	of the structure of consciousness within a framework of category
   628	theory; for example, one application is the examination of categorical equivalence of color similarity structure at the fovea and
   629	the periphery.
   630	Previously, peripheral and central color vision have been
   631	claimed to be “essentially the same” (Tyler 2015; Haun et al. 2017)
   632	or not (Dennett 1991; Lau and Rosenthal 2011). Between these
   633	visual fields, objective resolutions are different; yet, when the size
   634	of objects are matched for the resolution, essential visual phenomenology, including color experience, seems “essentially the
   635	same” (Gordon and Abramov 1977; Hansen et al. 2009; Webster
   636	et al. 2010). How can we proceed to extract and quantify the
   637	sameness of obviously different objects?
   638	First, you might think of “isomorphism” between category C
   639	and E for Central and Entire visual fields, respectively (Remember
   640	the section on “What does it mean for objects to be the same in
   641	a category? Isomorphic objects in a category”). As we introduced
   642	in the section “Category of categories”, C and E are objects of Cat
   643	(category of categories), where arrows are functors. Isomorphism
   644	in Cat, which is also called “categorical isomorphism,” requires
   645	that, for functor F: C → E, there exists functor G: E → C such that
   646	1 C = F;G and 1E = G;F. This turns out to be a very strong requirement and it does not work in the case between C and E here. This
   647	is because functor G collapses E into its subset C; thus, F cannot
   648	recover the original E from C.
   649	On the other hand, what occurs if we consider the “isomorphism” instead of equality in 1 C = F;G and in 1E = G;F? “Isomorphism” here is nothing but an invertible natural transformation,
   650	which is called natural equivalence, in functor categories Fun(C,
   651	C) and Fun (E, E). In this case, we obtain a weaker, yet very
   652	proper and flexible, kind of equivalence. In this sense, two categories C and E are equivalent. This is the level of equivalence
   653	that we are looking for, and potentially useful for consciousness research. The sameness, obtained through natural equivalence, is called “categorical equivalence” between category C
   654	and E.
   655	Definition: Categories C and E are categorical isomorphic if there are
   656	functor F: C → E and functor G: E → C such that 1 C = F;G and 1E = G;F.
   657	Category C and E are categorically equivalent if there are invertible
   658	natural transformations from 1C to F;G and 1E to G;F
       
   659	8
       
   660	Tsuchiya and Saigo
       
   661	Explaining the Yoneda lemma
   662	Finally, we are ready to introduce one of the most important
   663	results of category theory, the Yoneda lemma, to consciousness
   664	research. If we can apply the Yoneda lemma to consciousness
   665	research, we can characterize consciousness through its relationships with the others even if we cannot describe what consciousness is per se. To be more precise, the general conclusion of the
   666	Yoneda lemma is that the characterization of an object in a category is determined up to isomorphism by its arrows to the other
   667	objects in that category. We believe this is a substantial change of
   668	perspective, especially in the context of consciousness research:
   669	properties of an object are essentially the same as how the object
   670	relates with the others.
   671	Having said that, some may say that a concise and general
   672	direct definition of an object is vastly superior to the exhaustive
   673	characterizations of its relationships to other objects. However,
   674	as we noted in problems in characterizing consciousness on its
   675	own, the difficulty in dealing with consciousness is its difficulty
   676	to describe it in itself. How can I examine my definition of “redness” at the fovea compared to my “redness” at the periphery? As
   677	a start, I can compare my “redness” at fovea to all other experiences, including “redness” at periphery. Next, I can do the same by
   678	comparing my “redness” at periphery to all other experiences. This
   679	is an empirical approach to characterize a structural relationship.
   680	This procedure can establish equivalence of color experiences at
   681	the fovea and the periphery. The Yoneda lemma provides a mathematical footing to this empirical approach: we can eventually
   682	establish equivalence of my redness across the visual fields.
       
   683	Hom sets and hom functors
   684	Before applying the Yoneda lemma to consciousness categories,
   685	let us introduce only two more concepts: hom sets and hom
   686	functors.
   687	Definition: A collection of arrows from object X to object Y in category
   688	Q is called a hom set and written as homQ (X, Y). A hom functor maps
       
   689	a)
       
   690	b)
       
   691	c)
   692	hA(B) = homQ(A,B)
       
   693	Category Q
       
   694	a
       
   695	homQ(A,B)
       
   696	f
       
   697	hA(f) = homQ(A,f)
       
   698	a⨾f
       
   699	hA(C) = homQ(A,C)
   700	d)
   701	Hom functor
   702	h(A,-)
   703	- A set of arrows
   704	from A to all
   705	objects
       
   706	A
       
   707	A
       
   708	Hom functor
   709	h(-, A)
   710	- A set of arrows
   711	from all objects to
   712	A
       
   713	Figure 6. What are hom sets and hom functors? (a) A hom set, homQ (A,
   714	B), is a set of arrows from Apple (A) to Berry (B) in category Q. homQ (A,
   715	B) includes our familiar arrow, “nearly indistinguishable in color” but
   716	also many other different arrows (e.g. “indistinguishably tasty,”
   717	“indistinguishably sour,” etc). (b) A hom functor, hA, maps objects B and
   718	C in category Q into objects hA(B) and hA(C) in category Set. (c) hA also
   719	maps an arrow f in category Q into an arrow in category Set. (d) Visual
   720	depiction of hA: all sets of arrows emanating from A; and Ah: all sets of
   721	arrows pointing to A
       
   722	an object X in category Q into a set, which is an object in category Set. A
   723	hom functor also maps an arrow f in category Q into a function, which is
   724	an arrow in category Set. A function maps a set to another set in category
   725	Set (see Fig. 6a for hom sets and 6b and c for hom functors).
   726	Among hom functors, one that is obtained by fixing an object
   727	A in category Q is called hom functor hA. Let us go back to our
   728	familiar example: an object A for Apple. Then, hom functor hA
   729	can be understood as “all relationships that Apple has with other
   730	objects in category Q.” Formally, hA is a functor from category Q to
   731	category Set. This is because hA maps objects Berry (B) and Cherry
   732	(C) in category Q into homQ (A, B) and homQ (A, C) in category
   733	Set, which can be understood as “how Apple relates to Berry in
   734	all possible ways” and “how Apple relates to Cherry in all possible
   735	ways.” hA also maps an arrow f to homQ(A, f), which is an arrow
   736	in category Set. Note that an arrow in Set is a function from a set
   737	to another set. For unfamiliar readers, homQ(A, f) is a little confusing. To understand this, recall that f is one of the arrows from
   738	object A to B in category Q, which can mean “nearly indistinguishable,” for example. Let us single out one of the arrows in homQ
   739	(A, B) as an arrow “a,” which means A is “nearly indistinguishable
   740	in its color” with B. Then, a can be composed with f to obtain a;f.
   741	This compositional situation means as follows: A is related to C (in
   742	a sense of “a;f”), where C is related to B (in a sense of “f”), which is
   743	related to A as “nearly indistinguishable in its color”. Note that, in
   744	categories where there are more than one arrows between objects,
   745	an arrow from A to C, which means “nearly indistinguishable in
   746	its color,” may be a distinct arrow from a;f. hA(f) maps a particular arrow from A to B into the corresponding arrow from A to C.
   747	If you consider arrows as elements in a set, this means that hA(f)
   748	is a function between two sets of arrows, that is, homQ (A, B) and
   749	homQ (A, C).
   750	Note that until the last paragraph, we have always considered
   751	only one arrow from one object to the other, that is, we have simply considered categories of preorder in order to simplify various
   752	concepts in category theory. However, we will need to consider
   753	many arrows from one object to another object. homQ (A, B) is a
   754	powerful conceptual tool to think about such situations. We can
   755	consider homQ (A, B) as a list of properties of Berry in terms of
   756	relationship from Apple.
       
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   758	In our example of Central and Entire visual fields (Fig. 5), C
   759	and E will not be “categorically isomorphic,” which implies oneto-one relation on object and arrows since a functor from E to
   760	C should collapse the multiple objects or arrows into the same
   761	object or arrow. On the other hand, C and E can be “categorically
   762	equivalent” since it allows such multiplicity up to isomorphisms.
   763	If this can be experimentally verified, then we can conclude
   764	that content structures are “essentially the same” across visual
   765	fields. If, however, something essential is lost within a part of
   766	the visual field, say, blindspot, scotoma (Ramachandran and
   767	Blakeslee 1998), or colorblindness in a quadrant (Gallant et al.
   768	2000), then, categorical equivalence would not hold any more.
   769	Empirically, we can possibly test the existence of categorical equivalence between color similarity structures as content categories between the fovea and the periphery. While
   770	color similarity structure has been extensively studied in psychophysics (Kuehni 2010), most experiments allowed subjects to move their eyes freely in an unlimited time. As a
   771	result, the structure of color experiences is understood better at the fovea than the periphery, resulting in disputes
   772	(Cohen et al. 2020; Cohen and Rubenstein 2020; Dennett 1991; Lau
   773	and Rosenthal 2011; Tyler 2015; Haun et al. 2017). We surmise that
   774	a weaker notion of categorical equivalence is much more likely to
   775	be applicable in various types of structures of consciousness and
   776	more fruitful to seek for than the other stronger notions of the
   777	sameness (e.g. categorical isomorphism).
       
   778	A relational approach to consciousness
       
   779	9
       
   780	Pushing this idea further, we can denote a list of properties of
   781	all objects in category Q from the perspective of Apple as homQ
   782	(A, -) = hA. We use “-” to denote the argument in which any object
   783	in category Q can be substituted. For example, homQ (A,-) contains homQ(A, B) (i.e. a set of arrows from A to B, listing the
   784	properties of Berry from Apple’s viewpoint) as well as homQ(A,
   785	C) (i.e. a set of arrows from A to C). Thus, hA can be interpreted
   786	as all sets of arrows emanating from A to characterize the viewpoint of Apple. The dual notion is homQ (-, A) = Ah, which can be
   787	interpreted as all sets of arrows pointing to Apple. In other words,
   788	Ah is a list of properties of Apple, from the viewpoints of each and
       
   789	What can we say from the Yoneda lemma?
   790	The Yoneda lemma says:
   791	For each object A of C, the natural transformations Nat (hA,
   792	F) ≡ Hom(Hom(A,-), F) from hA to F are in one-to-one correspondence
   793	with the elements of F(A), that is, Hom(Hom(A,-)F) ∼
   794	= F(A). Moreover,
   795	this isomorphism is natural in A and F when both sides are regarded as
   796	functors from C × Fun(C, Set) to Set.
   797	Due to the technicality, we will omit the proof of the Yoneda
   798	lemma here. The readers who grasped all concepts introduced so
   799	far can understand the proof in a standard textbook of category
   800	theory (Awodey 2010; Leinster 2014). The important theorem that
   801	follows from the Yoneda lemma is as follows:
   802	Theorem: For objects A and B in category Q, hA ∼
   803	= hB is a
   804	necessary and sufficient condition for A ∼
   805	= B.
   806	Here, A ∼
   807	= B means that A and B are isomorphic objects in category Q. hA ∼
   808	= hB means that there exists a natural equivalence
   809	between hA and hB. A natural equivalence is a natural transformation from functor hA to hB, which is invertible. Remember a
   810	functor category, which we introduced previously. Given that hA
   811	and hB are both functors from category Q to category Set, they
   812	are objects of Fun(Q, Set). Thus, “hA ∼
   813	= hB” means hA and hB
   814	are isomorphic objects in Fun(Q, Set). In other words, there is
   815	an invertible natural transformation (i.e. a natural equivalence)
   816	between them.
   817	Let us translate the theorem into English. hA (and hB) means
   818	all relationships that A has with all other objects in category Q.
   819	Thus, hA ∼
   820	= hB means that all relationships for A and those for
   821	B are “naturally convertible” to each other (note that hA and hB
   822	includes the relationship from A to B and B to A as well). If A and
   823	B are isomorphic objects in category Q, then it is relatively easy
   824	to prove hA ∼
   825	= hB. However, a nontrivial mathematical fact is that
   826	hA ∼
   827	= hB can prove A ∼
   828	= B. Even if it is impossible to directly compare A and B, we can make a conclusion about it by examining how
   829	A and B relate to others. This approach is akin to the approaches
   830	taken in other fields; when studying meaning in semantics, environments in ecology, and astronomical objects in cosmology. Of
   831	course, the exact and precise approach is taken in mathematics.
   832	Here, we are proposing its application for consciousness research.
   833	∼ hB is more difficult to test
   834	Some readers may think that hA =
   835	empirically than A ∼
   836	B.
   837	Surprisingly,
   838	however, there are many
   839	=
   840	cases where it is overwhelmingly easier to check hA ∼
   841	= hB than
   842	A∼
   843	= B. There is no shortage of examples in mathematics. Next, we
   844	consider it in the context of consciousness research.
       
   845	A simple application of the Yoneda lemma in
   846	consciousness research
   847	Let us apply the above theorem for category of contents Q. To gain
   848	some insights, we consider the Checkershadow illusion (Adelson
       
   849	Figure 7. Applying the Yoneda lemma to category Q. (a) Checkershadow
   850	Illusion. The two squares labeled A and B look “different” in brightness
   851	as in complementary squares in a checkerboard. (b) Removing the
   852	surrounding of A and B to obtain A* and B*, whose brightness looks the
   853	“same”
       
   854	1995) (Fig. 7a). In this “illusion,” square A looks very dark, while
   855	square B looks quite bright. In Fig. 7b, we show square A* and
   856	B* without any background, which makes it easy to see that the
   857	brightness of A* and B* are physically the same. The striking effect
   858	of this “illusion” does not reduce even if you try to cognitively convince yourself that it is just a 2D image or that the direction of the
   859	shadow in the image is contradictory to the lighting condition in
   860	your room. It exemplifies the fact that the subjective experience
   861	of brightness of an object strongly depends on the context of the
   862	objects.
   863	Are the brightness of A and B the same? Can we define the
   864	subjective brightness of A and B? If there are individual differences
   865	in how we perceive brightness, is there any way to systematically
   866	examine and characterize subjective brightness?
   867	Let us consider squares A and B in Fig. 7a from a perspective of
   868	category theory. Within category Q, A and B are not isomorphic.
   869	Why? Here, we consider objects as squares denoted by A, B, i, ii,
   870	iii, and iv and arrows as “nearly indistinguishable” as before.
   871	First, consider if there is an arrow from A to i. Due to the
   872	clear boundary between A and i, they look quite different; thus,
   873	there is no arrow. Meanwhile, there is an arrow from B to i partly
   874	due to the fact that the bottom right part of i is indeed the same
   875	as that of B. However, if you pay attention to the top left part
   876	of the i, it becomes clear that i has a gradation and that i and
   877	B are not that similar. Yet, the cylinder on the top right provides an explanation of this difference within i as an effect of
   878	shadow. Furthermore, the overall configuration of the checkerboard also facilitates the sameness between B and i. In the end,
   879	we conclude the existence of an arrow from B to i. At this point,
   880	there is already a difference between a collection of relationships
   881	between A and the other objects and those between B and the
   882	others; thus, A and B cannot be isomorphic. This prediction coincides with our subjective phenomenology of “difference” between
   883	A and B, which provides a preliminary support of our framework of
   884	category Q.
   885	What has been said above applies similarly to i, ii, iii, and iv.
   886	According to our framework, this collection of arrows created a
   887	conscious experience where A and B do not look the same. Meanwhile, the theorem predicts that A and B should look the same
   888	if we erase the sources that generate the discordances between
   889	arrows from A and arrows from B. And this is a more meaningful aspect of the theorem: if hA ∼
   890	= hB, then A ∼
   891	= B. The sources of
   892	discordance may include the green column and its shadow, the
   893	regularity of the checkerboard pattern, etc. Let us consider any
       
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   895	every object in category Q. These concepts are schematically represented in Fig. 6d. hA and Ah are likely to be useful to consider
   896	the issue of subjective viewpoints in consciousness research.
       
   897	10
       
   898	Tsuchiya and Saigo
       
   899	Discussion
   900	In this paper, we introduced categories of consciousness in several
   901	forms and showed that the Yoneda lemma can be applied to obtain
   902	novel perspectives and predictions on consciousness. While the
   903	exact and concise definition of consciousness remains difficult,
   904	its characterization through indirect characterization of its relationships to others using convergent methods (see, for example,
   905	(Velmans 2009)) is actually a valid way as guaranteed by category
   906	theory and, in particular, the Yoneda lemma.
   907	We can already use some tools from category theory to characterize some aspects of consciousness in terms of level and
   908	contents in relational terms, as we demonstrate with some simple
   909	toy models. As the research makes progress, we surmise that we
   910	can possibly characterize all aspects of consciousness in relational
   911	terms in principle. For other related proposals of a relational characterization of concepts, see Chalmers (1996), Edelman (1996),
   912	Goldstone et al. (2005), Kleiner (2020), Loorits (2014), Signorelli et al.
   913	(2021), and Fink et al. (2021).
   914	As a future prospect, we think it is critical to consider various
   915	categories of consciousness within a larger structure of category
   916	of consciousness categories. Categorical equivalence between two
   917	conscious contents categories clarifies “in what sense” these consciousnesses are essentially the same.
   918	The importance and novelty of our proposal here can be better
   919	appreciated by the following criticism on similarity judgment in
   920	consciousness research by Pautz (2019):
   921	“To see how Similarity-Congruence falls short, let us return to
   922	hypothetical case … some unfamiliar sentient organism, Karl.
       
   923	of experiences, E1, E2, and E3, such that E1 is more like E2
   924	than E3. But, as a simple point of logic, Similarity-Congruence
   925	is not logically strong enough to tell us precisely what those
   926	experiences are… it doesn’t tell us whether they are colour
   927	experiences of similar shades of red, or whether they are colour
   928	experiences of similar shades of green. In fact, it doesn’t tell
   929	us whether they are experiences of colour or experiences of
   930	(say) smell. That is, it doesn’t entail the specific, determinate
   931	qualitative contents of those experiences.”
       
   932	Our proposal, specifically with the Yoneda lemma, is all about
   933	the power of “a collection of arrows (or relationships)” that is necessary to determine the color experience of red or green, the smell
   934	of fish, the sound of a bell, etc. And the essential role of a collection
   935	of relations does make sense in light of the fact that there has been
   936	no clinical report of a patient who lost color of red but not others.
   937	When brain damage causes color blindness, the reported cases
   938	are all about the entire loss of color experience (and associated
   939	concepts) in specific visual location (Gallant et al. 2000) but not
   940	a particular color or range of colors.2 Similar things can be said
   941	about visual motion (Zihl et al. 1983). Loss of a single category of
   942	perception, such as faces (e.g. prosopagnosia) and objects, has
   943	been linked to some specific brain lesions (Milner and Goodale
   944	1995; Kanwisher and Yovel 2006) but rarely on a particular face
   945	[except for the loss of “familiar” faces and objects, called “Capgras
   946	Syndrome” (Ramachandran and Blakeslee 1998)]. On the other
   947	hand, loss of a single linguistic concept (i.e. forgetting) is common.
   948	Why can we lose a specific single concept in the linguistic case but
   949	not for the experiences of color or motion, which is bound to the
   950	retinal locations? While anatomical localizations and functional
   951	mapping in neuroscience can give us a hint, they do not address
   952	this theoretical question.
   953	We believe that one possible answer to this riddle is to do with
   954	the Yoneda lemma. We hypothesize that the essence of the color
   955	experiences resides in relations with other colors (which may or
   956	may not be perceived at a given time), that is, without the entire
   957	sets of relations, color experiences just cannot emerge. On the
   958	other hand, we hypothesize that the essence of linguistic conceptual experiences [if any (Kemmerer 2015; McClelland and Bayne
   959	2016)] is its relational structure that allows one concept to be
   960	missing from the web of relations. In retrospect, therefore, the
   961	essential role of a collection of relations in consciousness may
   962	be obvious (Palmer 1999). Yet, we are not aware if a collection
   963	of massive similarity ratings has been ever collected or considered in consciousness research. Replacing the characterization of
   964	conscious experiences with a collection of comparative descriptions with other experiences can have a mathematically solid
   965	foundation: the Yoneda lemma in category theory.
   966	Combining the perspective of the Yoneda lemma with quantitative theories of consciousness, such as IIT (Oizumi et al. 2014;
   967	Tononi et al. 2016), we should be able to establish a new research
   968	program into animal consciousness (Tsuchiya 2017). Pioneering
   969	works in monkeys have introduced highly creative behavioral
   970	experimental tasks, such as binocular rivalry (Leopold et al. 2003)
   971	and no-report paradigms (Wilke et al. 2009), well beyond simple button press reports, which poorly characterize consciousness
   972	even in humans [also see Matsuzawa (1985) and Matsuno et al.
   973	(2004)]. These animal researchers have already brought various
       
   974	You want to determine exactly what experiences Karl has…
   975	Karl is presented with three objects consecutively … three Tshapes such that T1 is more like T2 than T3. Then, given
   976	Similarity-Congruence, you can deduce that Karl has some trio
       
   977	2
   978	Note that most people with color deficiency do not report having loss of
   979	a particular red and/or green. In fact, previous studies suggest these colordeficient people may be experiencing unique color dimensions, as revealed by
   980	the similarity rating experiments (Bosten et al. 2005).
       
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   982	points in the display as a potential object. Here, we assume A has
   983	a “nearly indistinguishable” arrow with a set of objects. Note that
   984	this set of objects include A itself (as “identity”) and B. If B has the
   985	arrow with the same set of objects, then mathematically, from
   986	hA ∼
   987	= hB, we can prove A ∼
   988	= B.
   989	In fact, after painting squares i, ii, iii, and iv into a white color as
   990	in the background, one of the authors (H.S.) now feels that A and
   991	B look the same. The other author (N.T.) feels that A and B became
   992	much more similar than before but they are still distinguishable.
   993	By removing further contextual cues, as in Fig. 7b, A* and B* look
   994	the same to most people. At this point, some readers may find that
   995	the arrows that we have adopted so far, signifying “nearly indistinguishable,” are too coarse to be useful in this situation. To capture
   996	different degrees of perceived similarity, we believe that “enriched
   997	category” will be useful (Lawvere 1973; Leinster and Meckes 2017;
   998	Fong and Spivak 2019; Tsuchiya et al. 2021). The Yoneda lemma
   999	is also known to exist in the enriched category (Bonsangue et al.
  1000	1998); thus, our argument here still holds in the enriched category.
  1001	Note that we are not claiming that this is a novel explanation about the Checker shadow illusion. In fact, the application of
  1002	the Yoneda lemma is never surprising as long as we work on preorder sets, where there exists at most one arrow between objects.
  1003	Unfortunately, there have been very few empirical psychophysics
  1004	studies that examined multiple arrows between objects so far.
  1005	This is a highly promising avenue of future research.
  1006	In sum, the Yoneda lemma predicts that two objects A and B
  1007	should look the same if we eliminate the discordances of relationships between A and the others vs B and the others.
       
  1008	A relational approach to consciousness
       
  1009	Conclusion
  1010	In this paper, we intentionally limited most of our examples into
  1011	cases where up to one arrow can exist from one object to the other
  1012	(preorder category). However, it is easy to extend our category Q
  1013	to have multiple arrows between objects, such as “nearly indistinguishable in terms of X,” where X can be color, shape, size,
  1014	location, etc. The richer the arrows, the better the structure of
  1015	the category can be understood. In addition to what we introduced in this paper, there are many more tools in category theory,
  1016	which are likely to be illuminating in consciousness research.
  1017	Some of these conceptual tools will clarify complex theoretical
  1018	concepts about consciousness, which have been discussed by
  1019	philosophers and psychologists for long. Such conceptual clarification will inspire further theoretical and empirical research ideas
  1020	to come in the near future.
       
  1021	Acknowledgements
  1022	The authors thank Ariel Zeleznikow-Johnston and Steven Phillips
  1023	for their comments on the earlier version of this manuscript.
       
  1024	Funding
  1025	This work was supported by Australian Research Council (N.T.,
  1026	grant number DP180104128, DP180100396); National Health Medical Research Council (N.T., grant number APP1183280); Japan
  1027	Science and Technology Agency (H.S., grant number JPMJCR17N2);
  1028	Foundational Question Institute (N.T. and H.S.); and Japan Promotion Science, Grant-in-Aid for Transformative Research Areas (N.T.
  1029	and H.S.). This project/research was supported by grant number
  1030	FQXi-RFP-CPW-2017 from the Foundational Questions Institute
  1031	and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
  1032	Community Foundation.
       
  1033	Conflict of interest statement
  1034	None declared.
       
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