When I first heard about the Yoneda Lemma, I thought of the Jeweled Net of Indra, and Saussure's conception of meaning as a web of relations. A word evokes from the reader the distinct meaning from its subjective difference from all other words. This is actually how Large Language Models today derive their high dimensional vectors from their training, and use them to create their proxy of meaning.
Category theory thus formalizes a system which our innate language instinct produces in each person, by the time they're five.
"The “Yoneda perspective,” that objects are their relations, is forced on you by the math.
The full faithfulness of the Yoneda embedding Y: 𝒞 → [𝒞ᵒᵖ, Set] is that perspective, formalized. Faithfulness sounds godly, but it’s a technical term meaning injectivity-on-morphisms, which just means different maps stay visibly different, which just means no confusion allowed.
People argue about nonduality vs. duality, but in category theory, duality isn’t optional. Why?
Because it’s not exactly about “things” (per se), nor is it about “relationships” (per se). It’s actually about the parallels between things and their relationships.
Curt’s aside: “Things” in category theory are called objects. “Relationships” are called morphisms. There’s something which is “higher” than this, and that’s a “natural transformation.”
In some sense, you can think of a natural transformation as a “relationship” between relationships, but that would be circular, so you need another name. Categorists call it a 2-cell."
Curt and/or Emily.
I believe that the state of SUPERPOSITION is the most fundamental expression/model of reality (AND the illusion) possible. Could the above be the state of superposition?
PLEASE, I sincerely IMPLORE you to investigate the links I am providing. My triune Fundamental Model Of Reality recognizes the state of superposition ITSELF. The relationship between entangled particles IS THE REALITY. We have mistakenly labeled the reality as "nothing" and one of the states of superposition (the "positive", "+", "thing", "particle") as the reality.
notice the "mathematics" column, "0" being synonymous with "=".
Thought experiment.
If I ask you to draw "concave" you will necessarily also, inadvertently without intent, simultaneously draw "convex". The definition of "concave" is: "the interior of a curve." The definition of "convex" is: "the exterior of a curve." The concave is not causally related to the convex and the convex is not causally related to the concave. Both convex and concave are causally related to " CURVE". If one forgets the entire concept of "CURVE", then it APPEARS as though the convex and concave are causal. This is the heart of the problem of DUALITY ITSELF! The triune FMOR is exactly its title.
People speak of simplicity, symmetry, elegance. And when presented with it they disregard it. It's fascinating.
My Substack posts are revealing the resolution to the question: "Is mathematics discovered or invented?" Mathematicians will have a very difficult time with this. Everyone will, but especially mathematicians.
maybe I can be of assistance in helping to clarify its relationship to physics.
The Yoneda Lemma deep connections to physics, especially in modern theoretical physics, where categories, dualities, and functors provide the mathematical backbone for concepts like quantum field theory, gauge symmetry even spacetime structure
fields as Functors
In physics, especially quantum field theory (QFT), we assign data (like field values or states) to regions of spacetime.
This is exactly what a functor does.
A field theory can be modeled as a functor from a category of spacetime regions (or cobordisms) to a category like vector spaces, Hilbert spaces, or algebras.
So
\text{Field Theory: } \mathcal{C}^{\text{op}} \to \mathbf{Vect} \text{ or } \mathbf{Hilb}
And then Yoneda says:
Any field configuration (i.e., an element of the functor) can be fully understood as a natural transformation from the “probing” functor — the spacetime region — to the field functor .
This captures the idea that observables in a region define the entire field content — exactly like in local QFT.
Tannaka-Krein Duality and Representation Theory
This is a direct application of Yoneda in physics
• Tannaka duality tells you that a group (or quantum group) can be reconstructed from its category of representations.
In physics, symmetries are central — particles and fields are classified by representations of symmetry groups.
Yoneda here provides the machinery to say:
The symmetry group is encoded in the natural transformations between representation functors.
So the group is reconstructed from how it acts — a very physical principle
You are what you do, not what you are internally.
Categories as Spacetime (in Topological QFT)
In topological quantum field theory (TQFT):
You model spacetime as a category of cobordisms (manifolds with boundaries).
The physical theory is a functor assigning vector spaces or amplitudes to these manifolds.
Yoneda tells you
Any way of assigning data to spacetime regions must respect the way they glue together — and this is naturally captured via transformations from representable functors (spacetime regions) to the physical theory functor.
Thus, the entire structure of a theory can be built from understanding how it behaves with respect to basic “probe regions.
Quantum Observables as Natural Transformations
In physics, observables are not absolute—they are relational. Yoneda’s Lemma encodes this
An observable or measurement is a natural transformation from a representable functor (probing an object) to a functor that assigns possible outcomes or states.
Just as a particle's identity is encoded in how it interacts, a physical system is encoded in its morphisms.
In other words:
There is no absolute system. Only relationships.
This is the core of relational quantum mechanics — and Yoneda formalizes it.
Derived and Higher Categories in String Theory and M-theory
In string theory, branes, categories of D-modules, derived categories, and stacks all rely heavily on the categorical language.
Yoneda gives the foundation for
Treating spaces not as point-sets but as functors (i.e., via functor of points).
Interpreting objects in terms of their external relationships, which is crucial when spacetime itself becomes "quantized" or emergent.
Yoneda Lemma becomes a deep principle:
You don't need to know the internal structure of a physical object to know what it is.
Knowing how it relates to everything else is enough — and in fact, is the object.
"People argue about nonduality vs. duality, but in category theory, duality isn’t optional."
I think this is misunderstanding the point of non-duality. The name is a bit confusing since you could posit non-dual versus dual, but deep non-dual insight transcends even this duality.
Buddhism talks about dependent origination and emptiness of inherent existence. This matches with the Yoneda lemma. Each object is empty of inherent existence, it does not exist on its own but always as the relation to the rest of the world. For "east" to exist, there has to be "west" as a contrast. Hence all things originate dependently on each other.
Been ringing this bell since learning about Yoneda Lemma & the Lakota greeting/introduction of Mitákuye Oyás’iŋ (All Are Related). Western academia is approaching indigenous/ancient wisdom and typically without acknowledging it.
Fascinating reflections; the conceptual insight calls to mind the works of Deleuze with striking parallels. I also wonder if anyone has considered potential connections in philosophy vis-à-vis the notion of "Cambridge Properties".
Getting into the deep waters — Yoneda, TQFT, quantum gravity, and holography come together in a way that is both conceptually elegant and foundational. The bridge.
Gravity as a Topological or Geometric Field Theory
Quantum gravity, at its core, seeks a theory of spacetime itself as a quantum object.
In many models — especially topological or background-independent approaches — we
Don’t fix spacetime in advance
Study amplitudes assigned to boundaries
Let the interior (the geometry “in between”) be summed over — a path integral over geometries
This is exactly the setting where TQFT ideas become relevant.
This can be seen as a natural transformation between hom-functors associated with the boundaries.
Yoneda then says
Every such transformation encodes a state of the gravitational field, as viewed through boundary probes.
So the “graviton” is not an isolated particle — it’s a relation between boundaries, reconstructed entirely from how regions transform into one another.
Higher Categories and Extended TQFT = Quantum Geometry
In extended TQFT and higher categories, we model not just
Points (0D)
Lines (1D)
Surfaces (2D)
Volumes (3D)
but their morphisms between morphisms — i.e., n-morphisms.
This matches quantum gravity’s need to
Treat points and regions relationally
Understand how different spacetime patches compose
Include quantum topologies or fluctuating geometries
The Cobordism Hypothesis (Lurie, Baez-Dolan) says
A fully extended TQFT is determined by its value on a point, up to dualizability data.
That’s the categorical version of a holographic principle
The entire tower of physical structure flows from the bottom — from the identity of the point and its interactions.
Yoneda as Ontology of Quantum Gravity
Let’s go deep
What is spacetime in quantum gravity?
Not a fixed arena, but a network of relations.
This is exactly what Yoneda formalizes.
Objects (regions of spacetime) are not primitive.
Their relations (cobordisms) and their interactions (natural transformations) define what they are.
The whole quantum spacetime is a category, and possibly a higher category, whose structure is reconstructed from how its pieces glue, evolve, and transform.
This is the relational view of reality — famously endorsed by Rovelli and others — and Yoneda gives it teeth.
To Summarize, Yoneda + TQFT → Quantum Gravity & Holography
Where in the natural world have you ever had an item (1). Placed nothing (0) next to it and that single item becomes ten (10).
This is the basis of mathematics. How can the basis of a language that cannot be seen in the natural world ever be taken seriously that it can explain the natural world?
Religion gave us gaslit ideas to gain power and control. Mathematicians are now doing the same thing.
If you truly wish to understand the natural world then use your emotions. Synthetic ideas based on our current knowledge fashion is not a good system.
It is another example of the human being not being exceptional but demented.
If we posit reality is made of something fundamentally, then we would expect that everything can be derived from that fundamental thing. Its nice to have a mathematical model to give some specifics to this fundamental, but how does a universe emerge from it?
Advaita Vedanta is a Hindu philosophical school that emphasizes non-duality, asserting that the individual self (Atman) is ultimately identical with the ultimate reality (Brahman) and that the perceived world is an illusion (Maya)
In the simplest framework, the mechanism of absolute dualism is the “not” function. It is the basis of all paradoxes found in self-circular linguistic and mathematical arguments and the relationship between quantum and classical structures. The format is that segments have the “not” function attached to the property they share. They are not members of each other even though they share a common named property. The “Not” Function—Paradox’s Mechanism in Linguistics, Mathematics, and Physics
I would like to see comments on my article " The Evolution of a Dual Universe and the Implications of Quantum Computing" I also believe that " Humans have the ability to take simplicity to the highest level of complexity"
Just for you Curt. If you don't want to click on it here you can get it in my substack chat. It's a reformulation of my graviton theory into Yonida Lemma language.
You are a developing trinity subjected to the pull of many dualities. Choose your spiritual major or minor in butt licking "me too" mediocrity or aspire to meet God as a worthy new member of the Divine Council
When I first heard about the Yoneda Lemma, I thought of the Jeweled Net of Indra, and Saussure's conception of meaning as a web of relations. A word evokes from the reader the distinct meaning from its subjective difference from all other words. This is actually how Large Language Models today derive their high dimensional vectors from their training, and use them to create their proxy of meaning.
Category theory thus formalizes a system which our innate language instinct produces in each person, by the time they're five.
"The “Yoneda perspective,” that objects are their relations, is forced on you by the math.
The full faithfulness of the Yoneda embedding Y: 𝒞 → [𝒞ᵒᵖ, Set] is that perspective, formalized. Faithfulness sounds godly, but it’s a technical term meaning injectivity-on-morphisms, which just means different maps stay visibly different, which just means no confusion allowed.
People argue about nonduality vs. duality, but in category theory, duality isn’t optional. Why?
Because it’s not exactly about “things” (per se), nor is it about “relationships” (per se). It’s actually about the parallels between things and their relationships.
Curt’s aside: “Things” in category theory are called objects. “Relationships” are called morphisms. There’s something which is “higher” than this, and that’s a “natural transformation.”
In some sense, you can think of a natural transformation as a “relationship” between relationships, but that would be circular, so you need another name. Categorists call it a 2-cell."
Curt and/or Emily.
I believe that the state of SUPERPOSITION is the most fundamental expression/model of reality (AND the illusion) possible. Could the above be the state of superposition?
PLEASE, I sincerely IMPLORE you to investigate the links I am providing. My triune Fundamental Model Of Reality recognizes the state of superposition ITSELF. The relationship between entangled particles IS THE REALITY. We have mistakenly labeled the reality as "nothing" and one of the states of superposition (the "positive", "+", "thing", "particle") as the reality.
https://www.nonconceptuality.org/1-fundamental-model-of-reality
notice the "mathematics" column, "0" being synonymous with "=".
Thought experiment.
If I ask you to draw "concave" you will necessarily also, inadvertently without intent, simultaneously draw "convex". The definition of "concave" is: "the interior of a curve." The definition of "convex" is: "the exterior of a curve." The concave is not causally related to the convex and the convex is not causally related to the concave. Both convex and concave are causally related to " CURVE". If one forgets the entire concept of "CURVE", then it APPEARS as though the convex and concave are causal. This is the heart of the problem of DUALITY ITSELF! The triune FMOR is exactly its title.
People speak of simplicity, symmetry, elegance. And when presented with it they disregard it. It's fascinating.
My Substack posts are revealing the resolution to the question: "Is mathematics discovered or invented?" Mathematicians will have a very difficult time with this. Everyone will, but especially mathematicians.
You must take what I am presenting seriously.
maybe I can be of assistance in helping to clarify its relationship to physics.
The Yoneda Lemma deep connections to physics, especially in modern theoretical physics, where categories, dualities, and functors provide the mathematical backbone for concepts like quantum field theory, gauge symmetry even spacetime structure
fields as Functors
In physics, especially quantum field theory (QFT), we assign data (like field values or states) to regions of spacetime.
This is exactly what a functor does.
A field theory can be modeled as a functor from a category of spacetime regions (or cobordisms) to a category like vector spaces, Hilbert spaces, or algebras.
So
\text{Field Theory: } \mathcal{C}^{\text{op}} \to \mathbf{Vect} \text{ or } \mathbf{Hilb}
And then Yoneda says:
Any field configuration (i.e., an element of the functor) can be fully understood as a natural transformation from the “probing” functor — the spacetime region — to the field functor .
This captures the idea that observables in a region define the entire field content — exactly like in local QFT.
Tannaka-Krein Duality and Representation Theory
This is a direct application of Yoneda in physics
• Tannaka duality tells you that a group (or quantum group) can be reconstructed from its category of representations.
In physics, symmetries are central — particles and fields are classified by representations of symmetry groups.
Yoneda here provides the machinery to say:
The symmetry group is encoded in the natural transformations between representation functors.
So the group is reconstructed from how it acts — a very physical principle
You are what you do, not what you are internally.
Categories as Spacetime (in Topological QFT)
In topological quantum field theory (TQFT):
You model spacetime as a category of cobordisms (manifolds with boundaries).
The physical theory is a functor assigning vector spaces or amplitudes to these manifolds.
Yoneda tells you
Any way of assigning data to spacetime regions must respect the way they glue together — and this is naturally captured via transformations from representable functors (spacetime regions) to the physical theory functor.
Thus, the entire structure of a theory can be built from understanding how it behaves with respect to basic “probe regions.
Quantum Observables as Natural Transformations
In physics, observables are not absolute—they are relational. Yoneda’s Lemma encodes this
An observable or measurement is a natural transformation from a representable functor (probing an object) to a functor that assigns possible outcomes or states.
Just as a particle's identity is encoded in how it interacts, a physical system is encoded in its morphisms.
In other words:
There is no absolute system. Only relationships.
This is the core of relational quantum mechanics — and Yoneda formalizes it.
Derived and Higher Categories in String Theory and M-theory
In string theory, branes, categories of D-modules, derived categories, and stacks all rely heavily on the categorical language.
Yoneda gives the foundation for
Treating spaces not as point-sets but as functors (i.e., via functor of points).
Interpreting objects in terms of their external relationships, which is crucial when spacetime itself becomes "quantized" or emergent.
Yoneda Lemma becomes a deep principle:
You don't need to know the internal structure of a physical object to know what it is.
Knowing how it relates to everything else is enough — and in fact, is the object.
Reminds me of Alfred North Whitehead's process thought.
"People argue about nonduality vs. duality, but in category theory, duality isn’t optional."
I think this is misunderstanding the point of non-duality. The name is a bit confusing since you could posit non-dual versus dual, but deep non-dual insight transcends even this duality.
Buddhism talks about dependent origination and emptiness of inherent existence. This matches with the Yoneda lemma. Each object is empty of inherent existence, it does not exist on its own but always as the relation to the rest of the world. For "east" to exist, there has to be "west" as a contrast. Hence all things originate dependently on each other.
I tried showing how this spiritual insight can be unified with science in this post: https://hiveism.substack.com/p/groundless-emergent-multiverse-20
Been ringing this bell since learning about Yoneda Lemma & the Lakota greeting/introduction of Mitákuye Oyás’iŋ (All Are Related). Western academia is approaching indigenous/ancient wisdom and typically without acknowledging it.
Fascinating reflections; the conceptual insight calls to mind the works of Deleuze with striking parallels. I also wonder if anyone has considered potential connections in philosophy vis-à-vis the notion of "Cambridge Properties".
Getting into the deep waters — Yoneda, TQFT, quantum gravity, and holography come together in a way that is both conceptually elegant and foundational. The bridge.
Gravity as a Topological or Geometric Field Theory
Quantum gravity, at its core, seeks a theory of spacetime itself as a quantum object.
In many models — especially topological or background-independent approaches — we
Don’t fix spacetime in advance
Study amplitudes assigned to boundaries
Let the interior (the geometry “in between”) be summed over — a path integral over geometries
This is exactly the setting where TQFT ideas become relevant.
Holography = Functorial Boundary-Bulk Correspondence
In the AdS/CFT correspondence, the central principle is The physics in the bulk is fully encoded on the boundary.
That sounds mysterious, until you recall In category theory, Yoneda tells you the entire object is determined by its morphisms to others.
And in TQFT
The bulk cobordism is determined by the linear maps between boundary states.
This sets up the holographic dictionary
Boundary = category objects (states, Hilbert spaces)
Bulk = morphisms between them (cobordisms, amplitudes)
Full theory = functor assigning these consistently
So the entire gravitational theory in the bulk is just a structured set of morphisms between boundary data — i.e., a natural transformation landscape.
Holography is Yoneda, physically realized.
Path Integral as a Natural Transformation
In gravity, the path integral assigns an amplitude to a spacetime region
Z(M): \mathcal{H}_{\text{in}} \to \mathcal{H}_{\text{out}}
This can be seen as a natural transformation between hom-functors associated with the boundaries.
Yoneda then says
Every such transformation encodes a state of the gravitational field, as viewed through boundary probes.
So the “graviton” is not an isolated particle — it’s a relation between boundaries, reconstructed entirely from how regions transform into one another.
Higher Categories and Extended TQFT = Quantum Geometry
In extended TQFT and higher categories, we model not just
Points (0D)
Lines (1D)
Surfaces (2D)
Volumes (3D)
but their morphisms between morphisms — i.e., n-morphisms.
This matches quantum gravity’s need to
Treat points and regions relationally
Understand how different spacetime patches compose
Include quantum topologies or fluctuating geometries
The Cobordism Hypothesis (Lurie, Baez-Dolan) says
A fully extended TQFT is determined by its value on a point, up to dualizability data.
That’s the categorical version of a holographic principle
The entire tower of physical structure flows from the bottom — from the identity of the point and its interactions.
Yoneda as Ontology of Quantum Gravity
Let’s go deep
What is spacetime in quantum gravity?
Not a fixed arena, but a network of relations.
This is exactly what Yoneda formalizes.
Objects (regions of spacetime) are not primitive.
Their relations (cobordisms) and their interactions (natural transformations) define what they are.
The whole quantum spacetime is a category, and possibly a higher category, whose structure is reconstructed from how its pieces glue, evolve, and transform.
This is the relational view of reality — famously endorsed by Rovelli and others — and Yoneda gives it teeth.
To Summarize, Yoneda + TQFT → Quantum Gravity & Holography
Where in the natural world have you ever had an item (1). Placed nothing (0) next to it and that single item becomes ten (10).
This is the basis of mathematics. How can the basis of a language that cannot be seen in the natural world ever be taken seriously that it can explain the natural world?
Religion gave us gaslit ideas to gain power and control. Mathematicians are now doing the same thing.
If you truly wish to understand the natural world then use your emotions. Synthetic ideas based on our current knowledge fashion is not a good system.
It is another example of the human being not being exceptional but demented.
If we posit reality is made of something fundamentally, then we would expect that everything can be derived from that fundamental thing. Its nice to have a mathematical model to give some specifics to this fundamental, but how does a universe emerge from it?
Advaita Vedanta is a Hindu philosophical school that emphasizes non-duality, asserting that the individual self (Atman) is ultimately identical with the ultimate reality (Brahman) and that the perceived world is an illusion (Maya)
In the simplest framework, the mechanism of absolute dualism is the “not” function. It is the basis of all paradoxes found in self-circular linguistic and mathematical arguments and the relationship between quantum and classical structures. The format is that segments have the “not” function attached to the property they share. They are not members of each other even though they share a common named property. The “Not” Function—Paradox’s Mechanism in Linguistics, Mathematics, and Physics
https://doi.org/10.31219/osf.io/ks458_v3
I would like to see comments on my article " The Evolution of a Dual Universe and the Implications of Quantum Computing" I also believe that " Humans have the ability to take simplicity to the highest level of complexity"
Just for you Curt. If you don't want to click on it here you can get it in my substack chat. It's a reformulation of my graviton theory into Yonida Lemma language.
https://drive.google.com/file/d/13ePC1RsD8MuYqkHgaxxbYU-_bnTLfVEb/view?usp=drivesdk
You are a developing trinity subjected to the pull of many dualities. Choose your spiritual major or minor in butt licking "me too" mediocrity or aspire to meet God as a worthy new member of the Divine Council