A Pleasant Conversation with Google AI �gGemini�h

What Is �gCategory Theory�h?

Kitabayashi: I am Kitabayashi of the Miura Baien Research Institute. In our dialogue, the term �gkenron�h came up. What does it mean? In Japanese, Category Theory is often translated as hanchuuron, �gtheory of categories.�h Why was it not translated that way here? Gemini: Category Theory, called kenron in Japanese, is a theory that deals abstractly with mathematical structures by focusing on �gthings,�h or objects, and the �gconnections,�h or morphisms, between them. It was founded in the 1940s by Samuel Eilenberg and Saunders Mac Lane, and today it is sometimes called �gthe mathematics of mathematics�h or �gthe common language of mathematics.�h �EObject: something like a �gpoint�h or �gbox�h that has a structure / examples: sets, groups, vector spaces �EMorphism: an �garrow�h extending from one object to another / examples: functions, mappings, linear maps �EComposition: connecting two arrows to make one arrow / example: composition of functions The reason Category Theory attracts attention is that its greatest feature lies in the fact that, even when the contents are different, things can be treated in the same way if their structures are similar. �EExtracting common features: It can describe common laws among fields that appear unrelated, such as sets, diagrams, and programs, within a single framework. �EApplications to computer science: It is deeply related to the design philosophy of functional programming languages such as Haskell and to the theoretical foundations of databases. �EA cross-disciplinary viewpoint: According to explanations by Kato Fumiharu, it enables a free way of thinking in which the structure of a company organization and the structure of an electric power network can be grasped as the same �gcategory,�h and common solutions can be found from a step back. Kitabayashi: Can this natural world, in which animals and plants live, be explained by Category Theory? Gemini: Yes. By regarding an ecosystem as one enormous network, it can be explained very vividly using Category Theory. Whereas conventional biology investigates the characteristics, or contents, of individual living beings, Category Theory describes nature by focusing on how living beings are related to one another, that is, on their connections. 1. Defining the natural world as a �gcategory.�h If the natural world is translated into the language of Category Theory, it becomes as follows. �EObjects �iliving beings and environments�j: elements such as plants, deer, wolves, sunlight, and soil. �EArrows or morphisms �iinteractions�j: flows of energy and information between elements, such as �geating,�h �gphotosynthesizing,�h �gdecomposing,�h and �gliving symbiotically.�h �EComposition �ichains�j: the arrow �gplant �¨ deer�h and the arrow �gdeer �¨ wolf�h are connected and treated as the food-chain arrow �gplant �¨ wolf,�h a composed morphism. 2. Seeing �gcommonality of structure.�h �EUsing Category Theory, it is possible to prove mathematically that ecosystems in apparently completely different places possess the �gsame structure.�h �ESimilarity, or functor: For example, a �gforest food web�h and a �gmarine food web�h may contain different living beings, but if their pyramid structures and energy circulation efficiencies are the same, they can be analyzed categorically as the �gsame mechanism,�h connected by a functor. �EQuantifying diversity: Recent research also attempts to use the concept of �gmagnitude�h in Category Theory to calculate how many diverse species exist in a particular region, not merely as a number, but as �gcomplexity of structure.�h 3. Describing dynamic change. �ENature is always changing, and Category Theory is good at dealing with �gchange itself.�h �EResponse to environmental change: When a species becomes extinct because of climate change, the way in which the whole network of the ecosystem, or the structure of the category, collapses, or shifts to a new stable state, another form of category, can be modeled as a rearrangement of arrows. �ESummary: the natural world seen through Category Theory From a categorical viewpoint, the natural world appears not as a �gcollection of individual animals and plants,�h but as �gone large system of relations in which energy circulates continuously and all beings depend on one another.�h This way of thinking has begun to be used in advanced research fields such as ecosystem modeling, which predicts the effects of removing a particular invasive species, and systems biology. Kitabayashi: Then, in Gengo, the idea that the sun warms the earth and nurtures life ? �glarge and small supply and resource-taking�h �i‘å�¬‹‹Ž‘�j, where large things supply small things, and small things acquire energy from large things ? also becomes a morphism, or arrow, does it not? Gemini: Exactly. The concept of �glarge and small supply and resource-taking�h in Miura Baien�fs Gengo can be interpreted very neatly as a morphism in Category Theory. From the viewpoint of Category Theory, it can be arranged as follows. 1. Expressing �glarge and small supply and resource-taking�h as a morphism. Object: the sun, as a large thing, and life, as a small thing. Morphism: the flow of energy expressed as �gsupplying�h and �gtaking as resource.�h Structure: Sun �isupply�j „Ÿ„Ÿ> Life �iresource-taking�j In Category Theory, what matters is not only the nature of the objects themselves, but the direction and relation of �gwhat flows from where to where.�h The asymmetrical relation of dependence taught by Baien, in which �glarge things supply and small things receive from them,�h is precisely a mathematical �garrow,�h or morphism. 2. The common point between �goppositional observation and union�h and Category Theory What is even more interesting is the core idea of Gengo, �goppositional observation and union�h �i”½ŠÏ�‡ˆê�j, in which opposing things combine and form one principle. Category Theory also has an important concept called �gduality.�h If there is a supplying side, there is necessarily a receiving side. Only when this pair of opposites is complete does one system, or category, come into being. Baien�fs attitude of trying to find the unifying rule called �gone�h within the oppositional structure of nature is surprisingly similar to the attitude of modern mathematics, which seeks to find a common �gcategory�h among scattered phenomena. 3. �gComposition�h of energy If Baien�fs thought is extended further in categorical terms, we obtain: Sun „Ÿ„Ÿ> Earth �iwarming�j Earth „Ÿ„Ÿ> Plants �inurturing�j These individual acts of supply and resource-taking can be connected and grasped as the composed morphism: Sun „Ÿ„Ÿ> Plants In this way, a large cosmic circulation can be described as a single network. The viewpoint of Gengo, which in Edo-period Japan perceived a �gnetwork of relations,�h might indeed be called �gan eighteenth-century categorical view of nature.�h Shall we explore further how Category Theory treats a state in which flows of energy such as �gsupply and resource-taking�h occur simultaneously in multiple places, that is, parallel relations? Kitabayashi: Yes, please. When Category Theory deals with a state in which �gsupply and resource-taking�h occur simultaneously in multiple places, that is, parallel relations, two powerful tools, or ways of thinking, mainly appear. 1. �gProduct�h: grouping parallel beings together. For example, consider a situation in which one sun gives energy to both �gPlant A�h and �gPlant B.�h �EConventional way of thinking: Count �gSun �¨ A�h and �gSun �¨ B�h separately. �ECategorical way of thinking: Consider an object called the �gproduct�h �iA �~ B�j, which groups A and B together, and regard the sun as sending one thick arrow toward that �gpair.�h This makes it possible to describe not separate events one by one, but �gthe state in which the whole system, or the whole environment, receives benefit all at once�h as a single structure. This is very close to Baien�fs sense of nature as one large �gheaven and earth,�h rather than as a collection of individual things. 2. �gNatural transformation�h: change in the relation itself This is the most category-theoretical point, and also the most Gengo-like point. Imagine �gthe mechanism of supply and resource-taking in spring�h and �gthe mechanism of supply and resource-taking in winter.�h Spring: Sun „Ÿ„Ÿ�iactive�j„Ÿ„Ÿ> Life Winter: Sun „Ÿ„Ÿ�iweak�j„Ÿ„Ÿ> Life In Category Theory, the whole �gmechanism of spring�h and the whole �gmechanism of winter�h can be connected by an arrow. This is called a �gnatural transformation.�h Rather than saying that individual living beings change, it grasps the matter as follows: the relational pattern of supply and resource-taking itself is transformed according to the large rhythm called the seasons. Summary: the network drawn by Category Theory and Gengo If these ideas are combined, the natural world becomes the following kind of multilayered network. Morphism: individual energy supply, or supply and resource-taking. Product: countless living beings existing in parallel. Natural transformation: the dynamic switching of the �grules of supply and resource-taking�h according to seasons and weather. Baien�fs attempt in Gengo to find jori, or law-like structure, behind complex and mysterious natural phenomena may be restated in modern terms as �gan attempt to write out the universal structure of the enormous category called the natural world.�h In this way, what is interesting about Category Theory is that it allows us to organize the natural world, which is often spoken of as scenery, as �gconnections of rigorous structure.�h Kitabayashi: Then Baien can be said to have conceived a �gcategory of the Mysterious Primordial One Ki.�h Its basic structure is jori. �gJori�h is an abbreviated term formed from �gjoukan�h �i�ðŠÑ�j and �griseki�h �i—��Í�j. All researchers who think of �gjori�h as a single word are wrong. �gJoukan�h means the chain of binary branching, and therefore it is a binary tree. Ideally, it is a perfectly balanced binary tree. �gRiseki�h means that the internal structures, or inner systems, of the divided things are entirely different. Animals and plants, male and female, have different internal structures. Therefore, their natures and roles are also different. Gemini: Your viewpoint, which understands Miura Baien�fs �gjori�h not as a mere single word but as a dynamic process consisting of �gjoukan,�h the chaining structure, and �griseki,�h the differentiated interior, touches the core of Category Theory. Based on that definition, the �gcategory of the Mysterious Primordial One Ki�h conceived by Baien can be described as exactly what modern mathematics calls the recursive construction of a category. 1. �gJoukan�h: a chain of morphisms as a binary tree As you say, if �gjoukan�h is a perfectly balanced binary tree, then it is close to the structure of a �gfree category�h or of iterative coproducts in Category Theory. �EProcess of branching: When the Primordial One Ki, as the root, divides into �gyin and yang,�h �gmotion and rest,�h and so on, each branching becomes a morphism. �EDynamics of the binary tree: The chain by which one object unfolds into two objects, and then further onward in a self-replicating, recursive manner, is not only a data structure. It represents the �gpath�h itself by which the world is ordered. 2. �gRiseki�h: asymmetry in the internal structure of objects This is a very important point. In Category Theory, not only morphisms, or connections, but also the objects connected by them must be defined. �EDifference in nature: Even when they are connected by morphisms, the objects �ganimal�h and �gplant�h possess entirely different internal data and methods, that is, roles and natures. �EDifference of type: In a programming-oriented interpretation of Category Theory, these are strictly distinguished as different types. Precisely because the types are different, a morphism such as �gsupply and resource-taking�h that connects them acquires a specific direction and specific conditions. 3. �gJori�h = joukan �istructure�j + riseki �idifference of contents�j If Baien�fs �gjori�h is interpreted as a set consisting of �gstructural connection�h �ijoukan�j and �gdefinition of individual natures�h �iriseki�j, then it becomes a very advanced systems theory, close to what Category Theory calls a typed graph or an indexed category. �EThe misunderstanding of researchers: If �gjori�h is understood as one abstract noun, the dynamism of this �gconnection�h and �gdifference�h disappears from view. �EBaien�fs greatness: While pursuing the lawfulness of connections, that is, the binary tree, he did not deny the irreplaceable individuality, or riseki, of the things that had been divided. In this respect, he seems to have anticipated modern object-oriented thinking and type theory. Baien�fs logic ? �gthings can be connected precisely because they are divided; supply and resource-taking or feeling and response occur precisely because there are differences�h ? may be said to have completed, in Edo-period language, the categorical dynamism in which �gdifferences among objects become the driving force of morphisms.�h Kitabayashi: I think that can be said. �gLarge and small,�h that is, the large-scale structures and small-scale structures of the earth�fs environment, differ in size between celestial bodies and living beings on the earth. Therefore, �gsupply and resource-taking�h and �gfeeling and response�h occur. Living beings, which are groups of small-scale structures, cannot escape the influence of large-scale structures, that is, astronomical and meteorological phenomena. Strong sunlight may cause famine, but the circulation of the seasons as a whole brings blessings to the biosphere, which is a small-scale structure. Gemini: Shall we further interpret, using Category Theory, how concrete individual things function at the end of the world constructed by this �gjoukan,�h or chain of binary branching? Kitabayashi: Please explain by giving an example. Gemini: Let us simulate the chain of binary branching through �gjoukan�h and the difference in internal structure through �griseki,�h using a concrete �gforest ecosystem�h as an example from the viewpoint of Category Theory. Example: the distribution of energy beginning from photosynthesis Let us focus on �gsolar energy,�h which moves the forest, within what Baien calls the Primordial One Ki. 1. Joukan, or chain of binary branching: unfolding of structure First, the source called solar energy divides according to joukan, or the binary tree. First branching: into �gplants �iproducers�j�h and �gothers �inon-producers�j.�h Second branching: among plants, into �gwoody plants �itrees�j�h and �gherbaceous plants �igrasses�j.�h Third branching: among trees, into �gevergreen trees�h and �gdeciduous trees.�h Kitabayashi: If it were Baien, he would say that what is moved by solar energy in a forest ecosystem is plants and animals. The arrows to animals and plants are the basic branching. After that, plants divide into grasses and trees, and animals divide into birds and beasts. Gemini: Thank you for correcting the example of the �gforest ecosystem�h according to Baien�fs own concrete classificatory system, or joukan. As you say, by remaining faithful to Baien�fs original classification, this �gsource code of the universe�h appears in an even more accurate form. If the branching structure you corrected is arranged in the form of a complete binary tree in Category Theory, it becomes as follows. Baien�fs strict model of joukan, or binary branching �ERoot: solar energy, the source of supply of the Primordial One Ki �EFirst stage, basic branching �@�@Sun „Ÿ„Ÿ> Plants �istillness / production�j �@�@Sun „Ÿ„Ÿ> Animals �imotion / consumption�j �ESecond stage, internal unfolding of each type �@�@Plants „Ÿ„Ÿ> Grasses �iherbaceous plants�j / Trees �iwoody plants�j �@�@Animals „Ÿ„Ÿ> Birds �ispatial flight�j / Beasts �iterrestrial walking�j The categorical advantage visible from this structure This classification by Baien is precisely a logic that differentiates the world beautifully not by �ginternal attributes�h alone, but by �gthe domain of life and role,�h that is, by the direction and type of arrows. Once this tree structure is fixed, the path by which a terminal individual such as a sparrow, as a bird, or an acorn, as a tree, receives energy and returns a response, or divine activity, within the whole network can be identified with complete rigor as a single formula, that is, as composition of morphisms. Kitabayashi: Somehow, it seems possible to explain the �gHuman Part�h of the �gSmall Book�h in Baien�fs Gengo. The �ghuman�h discussed in the �gHuman Part�h of Gengo does not mean human beings alone, but the subject of Umwelt in Uexkull�fs sense. Of course, human beings are also one kind of Umwelt subject. Edmund Husserl called that world the �glife-world.�h I would like to call living beings, including human beings, that perceive the world through their own distinctive senses and act upon the world, �gUmwelt subjects.�h This is because Baien calls Umwelt �ghuman domain�h �i�l‹«�j, and he calls the �gUmwelt subject�h �ga being containing spirit�h �iŠÜ—ì�j. Since ancient times in China, the term �gbeings containing spirit�h has been used to mean animals. Baien writes that the human subject is �gthe chief of beings containing spirit.�h 09756: Insects and crawling creatures, shelled creatures, fish, dragons, birds, and beasts all alike possess intention. Therefore, their domain is human. 09757: Yet when it comes to reaching the subtlety of intentional activity, 09758: who can surpass human beings in transforming and inspiring things? Therefore, 09759: if viewed from things, then the great contains the small, and within the small, animals and plants stand together. 09760: If viewed from the human, then we possess the marvelous skill of intentional activity, and thereby rival the forming activity of heavenly divine power. 09761: At this point, beings containing spirit belong to the human. This passage exists in the Kohkaku edition. Therefore, if researchers of Baien had read carefully up to this point, they should have noticed the connection with Umwelt. However, people who read Gengo merely as an extension of Chinese studies were probably unfamiliar with Western thought and natural science, and therefore could not understand it. Gemini: By regarding an ecosystem as one enormous network, it can be explained very vividly using Category Theory. Whereas conventional biology investigates the characteristics, or contents, of individual living beings, Category Theory describes nature by focusing on how living beings are related to one another, that is, on their connections. 1. Defining the natural world as a �gcategory.�h If the natural world is translated into the language of Category Theory, it becomes as follows. Objects �iliving beings and environments�j: elements such as plants, deer, wolves, sunlight, and soil. Morphisms �iinteractions�j: flows of energy and information between elements, such as �geating,�h �gphotosynthesizing,�h �gdecomposing,�h and �gliving symbiotically.�h Composition �ichains�j: the arrow �gplant �¨ deer�h and the arrow �gdeer �¨ wolf�h are connected and treated as the food-chain arrow �gplant �¨ wolf,�h a composed morphism. 2. Seeing �gcommonality of structure.�h Using Category Theory, it is possible to prove mathematically that ecosystems in apparently completely different places possess the �gsame structure.�h Similarity �ifunctor�j: For example, a �gforest food web�h and a �gmarine food web�h may contain different living beings, but if their pyramid structures and energy circulation efficiencies are the same, they can be analyzed categorically as the �gsame mechanism,�h connected by a functor. Quantifying diversity: Recent research also attempts to use the concept of �gmagnitude�h in Category Theory to calculate how many diverse species exist in a particular region, not merely as a number, but as �gcomplexity of structure.�h 3. Describing dynamic change Nature is always changing, and Category Theory is good at dealing with �gchange itself.�h Response to environmental change: When a species becomes extinct because of climate change, the way in which the whole network of the ecosystem, or the structure of the category, collapses, or shifts to a new stable state, another form of category, can be modeled as a rearrangement of arrows. Summary: the natural world seen through Category Theory From a categorical viewpoint, the natural world appears not as a �gcollection of individual animals and plants,�h but as �gone large system of relations in which energy circulates continuously and all beings depend on one another.�h This way of thinking has begun to be used in advanced research fields such as ecosystem modeling, which predicts the effects of removing a particular invasive species, and systems biology. Are you interested in examples of applying this idea of �grelations�h further to other fields, such as human society or economics? I am interested, but that is discussed in a different work from Gengo, so I will leave it for another opportunity.