The Yoneda Lemma - phase transition frame

Category theory applied

One of two posts using the same seed. This post uses the Phase Transitions frame to select and sequence threads. The second post uses the same seed but uses the Musical Transitions frame. This post is built on one of the key insights from category theory, Yoneda’s Lemma.¹

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Seed Text

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“mathematical objects are completely determined by their relationships to other objects”

“the properties of a mathematical object are more important than its definition”

An object is fully determined by the collection of its relationships with other objects.

The Yoneda Lemma proves that two distinct objects are structurally identical if and only if their relationships to every other object in the category are identical.

A mathematical object is any entity that serves as a noun in a mathematical universe, defined by how it interacts with other nouns via “arrows” (functions or transformations).

Depending on the category an “object” changes form entirely.

In set theory the objects are sets of things, such as the set of all integers, the set of real numbers or any finite set that has been chosen. The arrows are functions that map from one set to another. Sets are equal if there is a one to one mapping from the elements of one set to another set.

In topology the objects are shapes and spaces, a sphere, a donut, a flat sheet of paper. The arrows are transformations stretch or bend the shape without cutting it or gluing it together.

In linear algebra in the objects are vector spaces where you can scale and add arrows (the vectors) together. The vectors can be rotated, tilted, scaled and combined in different ways.

Groups are sets equipped with operations or rule systems. A concrete example is a Rubik’s Cube, where the object is the set of all possible turning combinations, governed by the rules of how those turns combine. The arrows are functions that map one rule system to another while perfectly preserving the internal structure of the operations.

For sequences the objects are the notes and the actions are selection and sequencing using a specific link type.

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Frame used: Phase Transitions

Free to use under Creative Commons CC BY-NC-SA 4.0

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Threads

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Thread 1 - Objects in Their Arrows

Considered together
The empty set and
Its potential²

-

Not null or
Not not null is
Not null

-

Zero the empty set
Or a pivot on which possibility
Is balanced

-

Any two points
Are always on
The same line

-

A method to discover
Transfinite primes that
Contain finites

-

Math built up
From relations will be
Mostly irrational³

-

The world is all that
Is other than the water
Rippling into light

-

Water sand or
Particle and wave
Left as it is

-

Water fills its form, has phases, eddies, makes waves

-

Light over water
Mingling wavelengths
Light in water

-

Light on water
Conforms to itself
A simple

-

Complete resting
State or stopping on each
Interruption

-

No healing without
Sleep more sleep deep
Blocks of time lost

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Thread 2 - Phase Drift

After washing the dishes
Cleaning the sink
Light steam

-

First morning
Cleaning the sink
Cracks in the porcelain

-

After washing the dishes
Cleaning the sink
Hot water on steel

-

Collapsed my lungs
Systems mobilized
Esophagus removed

-

Pink drainage
Craving water
Maybe salt cod

-

Piri piri between
Pain and irritation up
My right side

-

Rubbing skin
With a cold dry stone
Reduces irritation

-

The pain sighs from
Piri piri to hiri hiri as
Air touches the skin

-

The choices fan
Out in potential if I
Choose to continue

-

Imagining my bird
Breaths as I remember
My father’s⁴

-

We cant die
Each others death and
Even the roots rot

-

Pain saps strength
A required position
Of perfect stillness

-

Green breath as I work to recover
Blood oxygen
Remains low

-

Bathing in the breath
of all these plants is more
than restful

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Thread 3 - Snow Water

Water some snow
Then tea waiting for
The tannin’s colour

-

Rain sodden snow
Slumps off the cedar
A moment’s silence

-

Rendered opaque
By rain the snow
Slumps

-

The vague point where
Rain turns to snow
Snow to slush

-

Rain melted snow
Pooling on ice a dull
Fierce grey

-

Warmth as the snow
Melts into the streets and
Soaks the ground

-

Of water in
Air condensing
Into rain

-

The same water
Under different clouds
The same water

-

Each number clings
To its next like water
Beads on oil skin

-

Swallowing water
A gurgling in the
Stomach gullet

-

Something as simple
As drinking a glass of water
About to change

-

Soft water balancing
The stress the acid puts
On my digestion

-

The distance between
Breath repetition and our
Slow dying

-

Tea during rain
Stills time into a
Small brown space

-

Translation reveals
Both what can and cannot
Cross linguistic boundaries

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(like water in air)

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Thread 4 - Transitions in Care

New Year morning
Back from the hospital
Washing dishes

-

Yoshie home from the hospital
By the stove
Making okayu

-

After a long stay
In the hospital she is
Standing by the sink

-

The top of the wave
Travelling faster than the base
Spilling over

-

The Iribarren number
Angle of the slope over
The square root of wave height⁵

-

Times deepwater length
Spilling breakers
The wave steepens

-

Light on water
Conforms to itself
A simple

-

Wave object events
Dispersion into randomness
As it recedes

-

Kolmogorov
Measuring randomness
Of a sequence will not halt⁶

-

Home after travel
Drinking water and eating
Hard dry cheese

-

Morning peeling fruit
For her recovery the knife
Slightly dull

-

Picking thin bones
From the salmon as it grills
I scorch my fingers

-

The stone fits
The hand for
Different purposes

-

A weight remembering
Love resting on my
Ribs held still

-

Holding the stone
Remembering holding
Her hand

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Thread 5 - Emptiness

Not null or
Not not null is
Not null

-

And what empties
Itself of itself remains
Itself is empty

-

Of water in
Air condensing
Into rain

-

Not empty or
What is empty
Is not empty

-

And the self
That empties itself from itself
Is itself empty

-

In the air
Water condenses
Becoming rain

-

Set—ensemble
Set English for
Ensemble. A sequence.

-

Ensemble French for Set.
The whole together.
At the same time.

-

What part of a set are we?
What sets are we in?
We ask each Note.

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Emptiness not spontaneous
Empty of emptiness absorbs
All verbs and prepositions

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The present bubbling
Up in the constant change
Of cloud condensation

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Translation is not merely
Linguistic conversion but
Philosophical interpretation

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Thread 6 - Ascent

Bird song stops for
An interminable moment then
Starts again⁷

-

Safety in confined
Space trapped in a bubble
Of space time

-

A warm spot on
The floor where the
Dog was sleeping

-

Morning complete surround
Of bird song silence penetrating
Dark green leaves

-

Which bird is that
High intermittent complete
Maybe a skylark

-

Time always
Intermittent incomplete
In its passing

-

Passing through yellow
Passing through blue the
Green light touching grey

-

Bird song scattering
Across the top of the branches
As light slips through

-

A bird in clouds
Again this autumn
Glad I am older⁸

-

Again this autumn
I know I am older
Walking the shore

-

The world of dew
Under my feet a
Fall morning

-

And yet
(the world of dew)
We breathe together⁹

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Note: Each Note is treated as an “object” identified only by its pattern of incoming and outgoing links (closure, expectation, surprise, drift, etc.). The threads differ by which aspect of each Note’s relational profile is activated: sometimes a Note closes an argument, sometimes it becomes a hinge or a deceptive deflection, sometimes it acts as a quiet colimit of many small repetitions. I am trying a longer skein here, perhaps too long.

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Skein

Considered together
The empty set and
Its potential

-

Not null or
Not not null is
Not null

-

Zero the empty set
Or a pivot on which possibility
Is balanced

-

Set—ensemble
Set English for
Ensemble. A sequence.

-

Ensemble French for Set.
The whole together.
At the same time.

-

What part of a set are we?
What sets are we in?
We ask each Note.

-

Water some snow
Then tea waiting for
The tannins colour

-

Rain sodden snow
Slumps off the cedar
A moments silence

-

Rendered opaque
By rain the snow
Slumps

-

The vague point where
Rain turns to snow
Snow to slush

-

Rain melted snow
Pooling on ice a dull
Fierce grey

-

Warmth as the snow
Melts into the streets and
Soaks the ground

-

Of water in
Air condensing
Into rain

-

The same water
Under different clouds
The same water

-

Each number clings
To its next like water’s
Loose bonding

-

After washing the dishes
Cleaning the sink
Light steam

-

First morning
Cleaning the sink
Cracks in the porcelain

-

After washing the dishes
Cleaning the sink
Hot water on steel

-

New Year morning
Back from the hospital
Washing dishes

-

Yoshie is home from the hospital
By the stove
Making okayu

-

After a long stay
In the hospital she is
Standing by the sink

-

Collapsed my lungs
Systems mobilized
Esophagus removed

-

Pink drainage
Craving water
Maybe salt cod

-

Piri piri between
Pain and irritation up
My right side

-

Rubbing skin
With a cold dry stone
Reduces irritation

-

The pain sighs from
Piri piri to hiri hiri as
Air touches the skin

-

Pain saps strength
A required position
Of perfect stillness

-

Green breath as I work to recover
Blood oxygen
Remains low

-

Bird song stops for
An interminable moment then
Starts again

-

Morning complete surround
Of bird song silence penetrating
Dark green leaves

-

Which bird is that
High intermittent complete
Maybe a song sparrow

-

Again this autumn
I know I am older
Walking the shore

-

The world of dew
Under my feet a
Fall morning

-

And yet
(the world of dew)
We breathe together

-

Emptiness not spontaneous
Empty of emptiness absorbs
All verbs and prepositions

-

The present bubbling
Up in the constant change
Of cloud condensation

-

Translation is not merely
Linguistic conversion but
Philosophical interpretation

-

A Note is fully determined
By the collection of its
Relationships with other Notes

-

We are here and we share
That we are here
In this ensemble

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Some references

Cheng, Eugenia (2022) The Joy of Abstraction: An Exploration of Math, Category Theory, and Life. Cambridge University Press.

Long, Matthew (2026) “The Yoneda Lemma as Physical Law.” YonedaAI https://www.yonedaai.com/papers/yoneda-physical-content.html

Resnik, M. D. (1997) Mathematics as a Science of Patterns, Oxford University Press.

Yoneda Nobuo (1954) “On the homology theory of modules,” J. Fac. Sci. Univ. Tokyo Sect. I, vol. 7, pp. 193-227, 1954.

1. Category theory: A high-level branch of mathematics often described as the "mathematics of mathematics". It acts as a universal language that organizes abstract mathematical structures, focusing on relationships (arrows) and transformations between objects rather than the internal makeup of the objects themselves,

Instead of looking at what an object is (like the specific elements inside a set), category theory looks at how objects interact with one another. A category is made up of four foundational parts:

• Objects: The nodes or “things” in the system (e.g., all numbers, all geometric shapes, or all computer data types).

• Morphisms (Arrows): The relationships or transformations that connect one object to another (e.g., a function that turns one shape into another).

• Composition: The ability to chain these arrows together. If you have an arrow from A to B, and an arrow from B to C, you can compose them into a direct arrow from A to C.

• Identity: A default “do-nothing” arrow for every object that links it to itself.

Yoneda’s Lemma is described in the seed text. It is a mathematical formulation of the key Buddhist doctrie of ‘no independent origination.’ "No independent origination" means that nothing exists as a singular, independent entity. Everything arises solely in reliance upon a vast web of causes, conditions, and parts. If the conditions change or dissolve, the phenomenon itself changes or ceases.

2. The empty set or null set { } or a zero with a slash through it. The character phi Φ is often used for this though technically this is a mistake. 無 Mu. Way back when this kanji represented a person holding decorative feathers or ox-tail tassels in both hands. It represents a shaman or medium performing a ritual dance. The origin of all dependent things. Under standard Zermelo-Fraenkel set theory (ZFC), the empty set is the foundation from which all numbers are constructed. Mathematicians formally define 0 = Φ, 1 = {Φ}, 2={Φ{Φ}}.

The empty set and 無 Mu are both future seed texts.

3. Most numbers are irrational, there are infinitely more irrational numbers than rational.

4. Shortly before dying many people go into a phase of rapid, shallow breathing. This happended with my father 20 or 30 minutes before he died.

5. Irribaren number: a dimensionless ratio used to describe wave behavior as it approaches the shore. It is defined as the ratio of the slope's steepness to the square root of the wave's steepness.

Where Beta is the angle of the slope, H is the wave height, and L) is the wavelength.

6. Kolmogorov complexity (also called algorithmic information theory). Introduced in the 1960s, it defines the complexity of data as the length of the shortest possible computer program capable of generating that data.

7. I am mostly remembering a lark soraing up sining on the Arakakawa dote (levee).

8. Riffing on Basho.

この秋は 何で年寄る 雲に鳥

Kono aki wa / nande toshi yoru / tori ni kumo

This autumn
Why am I older
A bird into clouds

In my memory this has encoded as

この秋も 何で年寄る 雲に鳥

Kono aki mo / nande toshi yoru / tori ni kumo

Again this autumn
Why am I older
A bird into clouds

9. Riffing off Issa

露の世は露の世ながらさりながら

Tsuyu no yo wa / tsuyu no you nagara / sari nagara

The world of dew
Is the world of dew
And yet

The ‘world of dew’ is the world built on the empty set, or on the condensation of water vapor.