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Study Materials

Overview

• 维基百科: Yoneda lemma
• n-Lab: Yoneda Lemma (discussion)
• n-Lab: Yoneda Lemma

Expositions

• Lukas Heger. A Brief Introduction to Categories. Part 4. The Yoneda Lemma. With examples and exercises.
• The Yoneda Perspective
• Math Overflow: The Philosophical Meaning of the Yoneda Lemma
• Matthew Ward. The Brilliance of the Yoneda Lemma. A beautiful revolution of the concept of “space”.
• Eduardo Ochs: A skeleton for the proof of the Yoneda Lemma
• Eduardo Ochs: Notes on the Yoneda Lemma

Ideas

• Bartosz Milewski. On Composability.
• Towards a Science of Embodiment The Yoneda Lemma allows for a "homunculus" - Turing oracle.

Examples

• Math Overflow: Simple show cases for the Yoneda lemma
• Yoneda lemma as a generalisation of Cayley's theorem
• Proofsable: Yoneda Embedding Examples

Related Concepts

• 维基百科: Hom functor and internal Hom functor
• 维基百科: Closed category
• 维基百科: Representable functor
• 维基百科: Universal property

Readings

• Math Stack Exchange: Basic example of Yoneda lemma mentions subroutines.
• 维基百科: Continuation passing style: CPS transform is conceptually a Yoneda embedding.
• Bartosz Milewski. The Yoneda Lemma.
• Bartosz Milewski. Yoneda Embedding.
• What You Needa Know about Yoneda. Profunctor Optics and the Yoneda Lemma. (Functional Pearl) Guillaume Boisseau, Jeremy Gibbons. Mention the preorder example as proofs by indirect equality.
• Category theory notes 16: Yoneda lemma (Part 3) Linguist Chenchen (Julio) Song works through the details in his own way.
• Vladimir Voevodsky. 2016. Multiple Concepts of Equality in the New Foundations of Mathematics (slides)
• Voevodsky: Substitutional (definitional) equality. Transportational equality (witnessed by elements of the Martin-Lof identity types)(propositional).
• Quora: How is the Yoneda Lemma related to path induction?
• https://homotopytypetheory.org/2012/05/02/a-type-theoretical-yoneda-lemma/
• nLab: Space and Quantity: Yoneda Lemma and Generalized Spaces
• nLab: Geometry of Physics: Categories and Toposes: Yoneda Lemma and Generalized Spaces
• Daniel Murfet. The Yoneda Lemma
• Andreatta et al. Towards a Categorical Theory of Creativity for Music, Discourse, and Cognition based on the Yoneda lemma

Preliminaries

Categories

Categories

• In a category, if two objects have morphisms into each other, than they are isomorphic. So if we are only interested in nonisomorphic objects, then we can consider equivalence classes. And we can see that the maps only go in one way, so they are naturally partially ordered.

Objects

• An object splits a morphism into two. A morphism is expression (of content) and as such splits contradiction into two.
• When you have an object, then you ignore the other objects, the "non-objects", thus in the binomial theorem you don't have to deal with their relationships.

Isomorphism

• Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D.

Functors

• For any functor {$F$}, we have {$F_{id(X)}=id_{F(x)}$}. This is because for all morphisms {$h_{XX}$}:

{$$F(id_X)F(h_{XX}) = F(id_X h_{XX}) = F(h_{XX}) = F(h_{XX}id_X) = F(h_{XX})F(id_X)$$}

Functors

• Representable functors - based on arrows from the same object.
• Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces the object d to a set of the arrows from c to d. It reduces an arrow f:a to b to a set function from (c,a) to (c,b). Every functor is about forgetting, and the least forgetful functor is simply an isomorphism. But the category Set has very little explicit information, so in some sense it is the space of greatest forgetting, it is a blank slate.
• Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less.
• Functor is a contextualization. Is Z a group or an abelian group? "Form follows function".
• Functors preserve all paths.
• Embedding - fully faithful functor.
• Functors map morphisms.
• Functors can preserve or lose information but they never add any.
• Functors into {$\mathbf{Set}$} are special.
  • Sets allow us to invert functors. They let us go from determinism to nondeterminism. Functors are deterministic, they reduce information.
  • For example, given the functor which takes a determinant, we can have a functor that goes from the value of a determinant to the set of all matrices that have that determinant.
  • That's why the Axiom of Choice is so problematic. It introduces the notion of an outside perspective that chooses.

Natural transformations

• Natural transformations are the morphisms in a functor category.
• Objects in C are the indices for the components of a natural transformation in {$D^C$}.
• Morphisms in C are the indices for the commutative diagrams in {$D$} satisfied by a natural transformation in {$D^C$}.
• A commutative diagram in {$D$} satisfied by a natural transformation in {$D^C$} is a morphism equation in {$D$} that is validated by the morphism in {$D$} that the equation expresses.

Conceptual ideas

• Function composition elementwise mirrors (in the opposite direction) function composition setwise.
• Every object has a morphism to itself.
• Every set function has at least one element in its range.
• Functors take us from more refined (fewer relations, more equivalence classes) to less refined (more relations, fewer equivalence classes). This is counter to the representation of the twosome which takes us from "same" to "different".
• What is left unspoken: Sets are labeled - elements are labels. Categories are unlabelled only structure. Is this related to model theory?
• The properties of an entity correspond to the analogous stances: "This is the essence of the entity, the property that makes it what it is."
• A natural transformation equals a commutative diagram.

'Hom-sets and {$\mathrm{Hom}$} functors

Given an object {$A$} of {$C$}, define the category {$\mathrm{Hom}(A,*)$} in terms of {$C$} as follows:

• An object is, for some {$X$} of {$C$}, the set {$\mathrm{Hom}(A,X)$} of arrows {$A\overset{*}{\rightarrow}X$}. Note that the initial {$A$} is the same across the entire category, but the final {$X$} is specific to the object.
• An arrow is, for some {$g:X\rightarrow Y$} of {$C$}, a set function {$\mathrm{Hom}(A,–)(g)$} from {$\mathrm{Hom}(A,X)$} to {$\mathrm{Hom}(A,Y)$} which takes an element {$A\overset{h}{\rightarrow}X$} and sends it to the element {$A\overset{h}{\rightarrow}X\overset{g}{\rightarrow}Y$}.
• Given {$X\overset{f}\rightarrow Y$} and {$Y\overset{g}\rightarrow Z$} of {$C$}, composition of {$\mathrm{Hom}(A,–)(f)$} and {$\mathrm{Hom}(A,–)(g)$} yields the set function {$\mathrm{Hom}(A,–)(fg)$} which maps element {$A\overset{h}{\rightarrow}X$} to {$A\overset{h}{\rightarrow}X\overset{f}{\rightarrow}Y\overset{g}{\rightarrow}Z$} and thus clarifies {$g$} by {$f$}.

Extension extends a path with regard to what is outside. Clarification clarifies the inside of the path.

Given an object {$A$} of {$C$}, define the hom-functor {$h^A = \mathrm{Hom}(A,–)$} from {$C$} to {$\mathrm{Set}$}.

• An object {$X$} of {$C$} is sent to {$\mathrm{Hom}(A,X)$}.
• An arrow {$X\overset{f}{\rightarrow}Y$} is sent to the set function {$\mathrm{Hom}(A,–)(f)$}.
• The composition of {$X\overset{f}{\rightarrow}Y$} and {$Y\overset{g}{\rightarrow}Z$} to yield {$X\overset{gf}{\rightarrow}Z$} (where {$g$} extends {$f$}) is sent to the composition of set functions {$\mathrm{Hom}(A,–)(fg)$} which sends an arrow {$A\overset{h}{\rightarrow}X$} to the arrow {$A\overset{h}{\rightarrow}X\overset{f}{\rightarrow}Y\overset{g}{\rightarrow}Z$} (so that {$f$} clarifies {$g$}).

Thus the extension of the ends of paths in {$C$} is mapped to the clarification of the starts of paths in {$\mathrm{Hom}(A,*)$}. An example of this is working backwards in trying to prove a statement true or false.

This describes a sort of duality. If there Exists a path {$g$} that extends path {$f$}, then for All paths that end in {$g$} there will exist those that pass directly from {$f$}.

Consider, for example, natural numbers with the arrows given by addition. Let A be the Zero.

Observations

• The notation Hom(X,Y) seems to be ambiguous given that we're working with both C and Cop.
• Hom(A,–) gives the perspectives from A, including perspectives on perspectives.
• alpha theta Hom(f,_) takes as input the entire morphism: what value it goes to X and where that goes to in Y
• Natural transformation of Hom functors.

The functor {$\mathrm{Hom}(\_\;,\_\;)$} is a bifunctor. It can be written {$\{\_ \rightarrow_C \_\}$}. Given morphisms {$f:B\rightarrow B'$} and {$h:A'\rightarrow A$} it yields the set function {$\theta_{(f,h)}$} which sends {$g:A \rightarrow B$} to {$\theta(g)=f\circ g\circ h:A'\rightarrow B'$}.

{$\mathrm{Hom}$} functor acts on {$\mathbf{Set}^{C^{op}}$}

The Yoneda Embedding

{$H_\bullet:C\rightarrow [C^{op},\mathbf{Set}]$} sends {$A$} to {$H_A$} and sends {$f:A\rightarrow A'$} to {$H_f:H_A\rightarrow H_{A'}$}.

{$H_A = C(\_,A)$}

{$H_\bullet = C(\_\;,\bullet)$}

{$C(\_\;,\bullet)$} sends the object {$\bullet$} to the functor {$C(\_\;,\bullet)$} and sends the morphism {$f:\bullet \rightarrow \bullet '$} to the natural transformation {$C(\_\;,f):C(\_\;,\bullet)_\rightarrow C(\_\;,\bullet ')$}.

The Yoneda embedding is the functor {$\{\_\rightarrow_C \bullet\}$} in {$(\mathbf{Set}^{C^{op}})^C$} which sends the object {$A$} to the functor {$\{\_ \rightarrow_C A\}$} and sends the morphism {$A \overset{f}{\rightarrow} A'$} to the natural transformation {$\{\_ \rightarrow_C A\} \overset{\{\_ \rightarrow_C \; f\}}{\rightarrow} \{\_ \rightarrow_C A'\}$}.

The Yoneda embedding is the functor which takes a morphism {$f$} in {$C$} to the natural transformation that appends {$f$} to one Hom-set to get another Hom-set. It appends it in the reverse order.

The natural isomorphism

Validation of the evaluation functor {$F(A)$} by a set function is isomorphic to validation of the Hom bifunctor {$\{ \{\_ \rightarrow_C A\} \rightarrow_{\mathbf{Set}^{C^{op}}} F(\_)\}$} by a set function.

Yoneda Lemma - Statement

{$\mathrm{Hom}(B,C) \leftrightarrow \mathrm{Nat}((C\rightarrow\_),(B\rightarrow\_))$}

{$(\mathrm{How}\rightarrow\mathrm{What}) \rightarrow \mathrm{Nat}((\mathrm{What}\rightarrow\_),(\mathrm{How}\rightarrow\_))$}

Let {$F$} be an arbitrary functor from {$C$} to Set.

Yoneda's lemma says that: For each object {$A$} of {$C$}, the natural transformations from {$h^A$} to {$F$} are in one-to-one correspondence with the elements of {$F(A)$}. That is,

{${Nat} (h^{A},F)\cong F(A)$}

Moreover this isomorphism is natural in A and F when both sides are regarded as functors from SetC x C to Set.

The Left Hand Side

The left hand side {$L$} is a functor in {$\mathbf{Set}^{C^{op}\times \mathbf{Set}^{C^{op}}}$} which takes a morphism {$(f:A\rightarrow A',\alpha:F\rightarrow F')$}, applies the Yoneda embedding to {$f$}, applies the Hom functor {$\{\_ \rightarrow_{\mathbf{Set}^{C^{op}}} \_\}$} to the pair {$(\{\_ \rightarrow_C \; f\},\alpha)$}, and returns a set function {$\theta$}. In other words, it converts {$f$} to the natural transformation {$\{\_ \rightarrow_C A\} \overset{\{\_ \rightarrow_C \; f\}}{\rightarrow} \{\_ \rightarrow_C A'\}$}. Then it pairs that with the natural transformation {$\alpha$}. Together, by way of the functor {$\{\_ \rightarrow_{\mathbf{Set}^{C^{op}}} \_\}$}, they yield a set function {$\theta$} that maps a natural transformation {$\psi$} from the functor {$\{\_ \rightarrow_C A\}$} to the functor {$F(A)$} to the natural transformation {$\theta({\psi}) = \alpha \circ \theta \circ \{\_ \rightarrow_C \; f\}$} from the functor {$\{\_ \rightarrow_C A'\}$} to the functor {$F'(A')$}.

Note that this natural transformation {$\psi$} takes us from the Hom set (the How) to the arbitrary functor (the What). This means that any functor can be understood as the output of a Hom functor.

Meaning

Philosophical essence of Yoneda Lemma

• Consider the Yoneda Lemma as a duality between the conscious mind's questions and the unconscious mind's answers. The assembled questions and the assembled answers can be compared. I can compare this with my own sets of questions and sets of answers in my research. They may group themselves by divisions of everything.
• Tobler's First Law of Geography: Everything is related to everything else, but near things are more related than distant things. Relate to the Yoneda lemma.
• Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another.
• Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain.
• Consciousness - balance between synchrony and asynchrony.
• Synchronization. The chimera state is perhaps an example of the "popping out" of a subsystem.

Mathematical essence of Yoneda Lemma

• Riehl: Yoneda Lemma serves to classify natural transformations.
• Yoneda lemma: A set of actions becomes an object.
• Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f).
• Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. This relates a system and a pre-system. Thus it sets up the point where the pre-system becomes a system (a subsystem). And it can do this by distinguishing objects - those identity morphisms that do nothing - they get segregated into the subsystem and understood as such.
• Yoneda lemma - relates to exponentiation and logarithm.
• Executing an entire subroutine is in addition to executing individual steps. It is the use of a new parser at a second level.
• The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category.
• The crucial point of the Yoneda lemma is that the set function is on an object A and not on a morphism.

Identity morphism - null action

• The identity morphism falls out as a special morphism among the possible morphisms from B to B. It is the basis for Graziano's awareness schema.
• So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others.
• The do-nothing action indicates that the subsystem has its own independent time, independent system.
• Zero curvature relates two different ways of doing nothing. You go around a loop and you consider what happens under transport (by a connection) of a vector (whether it changes or not). Nonzero curvature is indicated if something happened.
• The null action establishes the equivalence of the actions before and after.
• "Does nothing" means that it has never has any effect on other actions upon composition with them. An action does something if it takes one action into another action. (Not whether it takes an object into another object.) Thus actions are maps on actions.
• Yoneda Lemma relates X=X where one X is unbounded (beyond system) and the other X is bounded (defined, in system). And the latter X is understood in three ways by the three-cycle.
• Yoneda lemma models determinism: If you know where the root of the tree goes, then you know where everything goes.
• The identity matrix amongst linear transformations.
• Presheaves: the restriction of a set {$U$} to itself.
• The identity permutation.

Form of Yoneda Lemma

• X<=Y <-> for all z(z<=x -> z<=y) Here we have semantic implication <= and logical implication -> And on the right hand side we have context, and in context we have a splitting of semantics and syntax. Syntax is relevant in context, where there is a language and a community for that.
• In logic, the difference between a statement (such as "you owe me money") being true and its being true and not empty is the difference between the natural numbers and the whole numbers. It's likewise the difference between weakly increasing and strictly increasing, less than or equal and less than. Weak increase is fundamental for the Yoneda Lemma so that a relation is reflexively satisfied by an object.

Two sides of the Yoneda Lemma

• Yoneda Lemma: Covariant version and contravariant version are two PDAs that come together to form a Turing machine.
• The Yoneda Lemma relates the global and local.
• The Yoneda lemma relates (contradictory) nontype theory and (noncontradictory) type theory. It embeds the noncontradictory into the contradictory, and also it interprets the contradictory in terms of the noncontradictory.
• Splits in two the validation, "division into two". The subroutine validates the changes in the internal step and the external step, but assumes that the incoming natural transformation has already been validated.
• Quantification. Yoneda Lemma for preorders works by making explicit the context (and quantifying over it).
• The left side of the Yoneda lemma relates the concepts of forgetful and free.
• Validating a single step, in a generic way, is equivalent to validating everything.
• The theta was validated by somebody else, and we just validate the extension.
• The extension happens on the inside and the outside, the validations balance each other, like left and right parentheses.
• In the Yoneda lemma, the natural transformation on the left hand side seems to coordinate vertical and horizontal composition.
• {$\textrm{Set}^X$} is the category for bundles over {$X$}
• Value vs. Variable
  • Verifying a calculation of values or of variables is the same.
  • Comparing components on values on {$A$} and {$A'$} vs. comparing application of natural transformation {$\alpha$} before or after the application of the Hom functor.
• Evaluate a function vs. Execute a step
• Functional programming vs. procedural programming
• Deterministic vs. Nondeterministic
  • P vs NP
  • coefficients vs. eigenvalues

Phase transition

• Randomness is related to symmetry breaking.
• The chance that an eigenvalue is 0 is related to the chance that a matrix is degenerate. And that chance is basically zero. And the chance that an eigenvalue does nothing is the chance that it is 1 and that chance is basically zero. Yet in a universe of possibilities that are related, there is a nonzero chance that there is a fixed point.
• Mass is an indicator of subsystems.

Check: This means that paths which start from A, composed by "clarification", are transformed into set functions that start on the element {$\eta(id_A) \in F(A)$} and proceed by "set function composition".

• The Yoneda Lemma asserts that {$C^{op}$} embeds in {${\textbf{Set}}^C$} as a full subcategory.
• In other words, every category embeds in a functor category.
• The functor category {$D^C$} has as objects the functors from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma describes representable functors in functor categories.

Observations

• F(A) is a set of labels.
• If F is the empty functor, or if F(A) is empty, then there are no natural transformations.
• The general case with many objects is not much more complicated. Again, the main idea is that the morphism {$g_{YX}$} is an element (in {$\mathrm{Hom}(X,Y)$}) and also thought of as a map (as {$\mathrm{Hom}(–,X)(g_{XY}^{op})$} from {$\mathrm{Hom}(X,X)$} to {$\mathrm{Hom}(Y,X)$}).
• alpha (f) <-> Hom (Hom(f,_),alpha)

Argument

Core of the argument

Given functors {$H:C\rightarrow\mathrm{Set}$} and {$F:C\rightarrow\mathrm{Set}$}, and natural transformation {$\eta:H\rightarrow F$}, then {$\eta_X:H(X)\rightarrow F(X)$} is a set function.

If the functor {$H=\mathrm{Hom}(X,–)$}, then the set {$H(X)=\mathrm{Hom}(X,X)$}. Any morphism {$h_{XX}:X\rightarrow X$} is inside this set: {$h_{XX}\in \mathrm{Hom}(X,X)$} (with composition by extension). But this morphism is also thought of outside as a set function {$\mathrm{Hom}(–,X)(h_{XX}):\mathrm{Hom}(X,X)\rightarrow \mathrm{Hom}(X,X)$} (with composition by clarification).

Consider what happens to the element {$id_X\in\mathrm{Hom}(X,X)$}. The following diagram must commute:

{$(F(h_{XX}))(\eta_X(id_X))=\eta_X(h_{XX}(id_X))$}

Composition by clarification deems that the set function {$h_{XX}(\mathrm{Hom}(X,X))$} outputs the subset of those morphisms in {$\mathrm{Hom}(X,X)$} which start with the morphism {$h_{XX}$} in {$C$}. In particular, this set function maps each morphism {$g_{XX}$} in {$C$} to the morphism {$h_{XX}g_{XX}$} in {$C$}.

Thus, in particular, the set function {$h_{XX}$} maps the elements {$h_{XX}(id_X)=h_{XX}$}. Here again the morphism {$h_{XX}$} is an element (in {$\mathrm{Hom}(X,X)$}) and also thought of as a map (as {$\mathrm{Hom}(–,X)(h_{XX})$} from {$\mathrm{Hom}(X,X)$} to itself). But then:

{$\eta_X(h_{XX})=(F(h_{XX}))(\eta_X(id_X))$}

This means that the values of the set function {$\eta_X$} are all determined by the element {$\eta_X(id_X)\in F(X)$}.

In particular, the identity morphism {$id_X$} is a distinct element of {$\mathrm{Hom}(X,X)$} with composition by extension, and an omnipresent subpath in {$\mathrm{Hom}(\mathrm{Hom}(X,X),\mathrm{Hom}(X,X))$} with composition by clarification. In the first case, it is not implicit, but most be noted explicitly. In the second case, it is always implicit. Consequently, because of the first case, it gets represented by a specific element {$\eta_X(id_X)$}, and because of the second case, it is implicit in every represented path {$\eta_X(h_{XX})$}. Thus that specific element is the initial path in every compositional chain of set functions.

Applications

Application: Cayley's theorem

This case with a single object {$E$} in {$C$} is what is needed for Cayley's theorem. Let the morphisms {$E\overset{g}{\rightarrow}E$} be the elements of a group. Note that the group need not be finite.

First of all, Cayley's theorem is given by the fact that {$h^E=\mathrm{Hom}(E,–)$} is a functor.

• The set {$\mathrm{Hom}(E,E)$} is the set of group elements, the morphisms.
• The definition of the functor {$\mathrm{Hom}(E,–)$} shows that when it maps morphisms, it maps the group's elements to set functions from {$\mathrm{Hom}(E,E)$} to itself.
• In general, set functions do not form a group. However, permutations can form a group.
• The first fact to note that the functor {$h^E$} preserves group structure: identity element, inverses, associativity, closure.
  • The functor maps the group's identity element {$e$} to the identity set function.
  • The fact that inverses exist means that each group element gets mapped to a set function with an inverse. But this means that set function must be surjective and injective, thus a bijection, in other words, a permutation.
  • The functor respects associativity and closure.
• The second fact to note is that the functor is injective. Suppose that the functor maps two group elements {$f$} and {$g$} to the same set function {$h^E(f)=h^E(g)$}. Then given a group element {$k{\in}\mathrm{Hom}(E,E)$}, we have {$fk=h^E(f)(k)=h^E(g)(k)=gk$}. But then {$f=fkk^{-1}=gkk^{-1}=g$}.
• Note here that it didn't matter which element {$k$} we used for this argument. The argument works for every {$k$}. That is an important fact that makes groups work and involves a set of conditions. The Yoneda Lemma expresses this internal fact about groups externally, from the categorial point of view, as a set of natural transformations. Thus the same set of conditions appears internally, in the group structure, as a set of equations that can be solved (dividing by k), and externally, in the categorical relations, as a set of actions that can be performed (multiplying by k).

Thus we conclude that {$h^E$} maps group elements to permutations, preserves the group structure, and is injective. Thus the group is embedded in the group of permutations on the elements of the group.

The Yoneda Lemma provides the following perspective on Cayley's theorem.

• Note that we have a natural transformation from the functor {$\mathrm{Hom}_C(–,*)$} to itself, and that the only object in C is {$*$}, thus all four corners of the commutative diagram are the same: {$\mathrm{Hom}_C(*,*)$}, which is the set of group elements. Then we can consider what happens to any element of that set.

Application: Permutations, Matrices, Automata

Consider the duality between permutations/numbers/sets and matrices/vectors/lists/finite-automata.

Permutations are a group. And a group can be thought of as a single object with many relations to itself given by its elements. Whereas a matrix is the diagram of a category and describes relations between many distinct objects.

There is a map between permutations and matrices. How do we think of permutations when the group is described in terms of a single object? It seems then the permutations are described by structure within the group? What does Cayley's theorem say? And the Yoneda lemma?

In the case of a finite automata, we can think of it as a set of states linked by steps where a functor then maps each step (each morphism) to a letter (or word) in an alphabet. That functor thus maps all of the objects to a single object, thus a group. And we look at those paths that start at a start object and end at a final object. (How might they relate to initial and terminal objects?)

A group can be thought of as a single object with the elements as morphisms from it and to it. The identity element is the identity morphism. The actions come in pairs, doing and undoing, yielding the identity morphism. (Under what circumstances is there a canonical way to divide the pairs into "positive" and "negative"?) What makes the actions distinct? A symmetry group requires two points of view, level and metalevel, by which things can be "the same" and yet also "different".

Automata

• Pushdown automata have a stack of priorities. In general, automata deal with concerns - rūpesčiai.
• Yoneda Lemma. The set function {$\theta \rightarrow \alpha \theta \textrm{Hom}(f,\_)$} is a rule for a pushdown automaton. The {$\alpha$} comes from the finite automaton (the input) and the {$\textrm{Hom}(f,\_)$} should describe the stack of memory. This all, on the left-hand side, is compared to a finite automaton on the right-hand side.
• taisyklė (rule) = sudūrimas (composition)

Notes

• One natural isomorphism validates naturality.
• One validates the unit - the initial state.
• One validates the counit - the terminal state.
• Differentiable maps are defined by precomposing and postcomposing with differentiable charts. Thus this brings to mind pushdown automata, the Yoneda lemma, and the arisal of a coordinate system for an emerging subsystem.

Yoneda Embedding https://www.math3ma.com/blog/the-yoneda-embedding

• vantage point: how
• how an object looks: what

Representability

• Invariant - community truth
• Absolute, subjective truth (God) vs. practical, objective truth (world - three-cycle) vs. community truth
• The Stacks Project - Opposite Categories and the Yoneda Lemma
• Guillaume Boisseau, Jeremy Gibbons. What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl) also a good video
• Profunctor, adjoints, Yoneda lemma
• Guillaume Boisseau, Jeremy Gibbons. What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl).
• Yoneda Lemma: You can factor validation of composition into the the most recent step (before and after) and everything else (before and after). The whole point of category theory is the validation of composition. Commutative diagrams express composition. Natural transformations express validation indexed by underlying objects. Adjunctions leverage associativity.
• Yoneda lemma is based on pre-sheaves. But it should be interpretable in terms of four levels of information about sheaves given by the four functors on sheaves that (as four of six Grothendieck operations) are the four representations related to the levels of knowledge: whether, what, how, why.
• Double Embedded Yoneda Lemma? {$\forall F (F(a)\rightarrow F(b)) \Leftrightarrow (a\rightarrow b)$} Implication left to right by setting {$F$} to the identity functor. Implication right to left by applying the functoriality of {$F$} on the morphism.
• How: Presheaf expresses "the sort of thing that happens in a space". Space is a site.
• https://www.cs.ox.ac.uk/jeremy.gibbons/publications/proyo.pdf Boisseau, Gibbons. What You Needa Know about Yoneda. Profunctor Optics and the Yoneda Lemma.
• http://pi.math.cornell.edu/~zbnorwood/ucla/files/repfunctors.pdf Zach Norwood. How to prove that a non-representable functor is not representable.]]
• Yoneda Lemma relates absolute perspective and relative perspective in the way that the principle of least action relates kinetic and potential energy.

Yoneda Lemma

• The Yoneda Lemma expresses whether our language includes its metalanguage, or whether it does not. Syntax (logic) and semantics (math) can be in the same language if we use objects as states and arrows as execution. If we only have arrows as execution, but no states, then our program is compiled, and we have no meta language because we have no use of objects as states.
• Yoneda Lemma https://www.classe.cornell.edu/spr/1999-09/msg0017972.html
• Does the Yoneda lemma mean that you can compile a program? so that it runs hardwired? in terms of impenetrable but valid subroutines?
• Level of knowledge in the Yoneda lemma: Homset expresses All (every, none) and Morphism expresses One (any, some).
• Does the foursome, as with the Yoneda lemma, establish four natural bases for automata?
• nLab: universal construction and its relation to the Yoneda Lemma

nLab: Subobject classifier explains how the Yoneda lemma can be used to get the subobject classifier.

• The subobject classifier always comes with the structure of an internal poset; that is, a relation {$\subseteq\, \hookrightarrow \Omega\times\Omega$} which is internally reflexive, antisymmetric, and transitive.
• {$\Omega$} is an internal Heyting algebra.
• The collection of subobjects of any object is an external poset.
• Yoneda lema - prasmingumas - kodėl: objekto turinys išsakomas ryšių nes objektas tėra savo raiška, o raiška reiškiasi ryšiais.

Ninja's Yoneda lemma. Coend calculus.

• The moments of Chebyshev are given by the Catalan numbers which count Dyck paths. How are they related to pushdown automata? and the Yoneda lemma? and the Mandelbrot set?

Milewski on Profunctors and Optics

• Related to the natural isomorphism in the Yoneda lemma
• End, wedge condition, natural transformation (finite automata) <-> coend, cowedge condition, composition (pushdown automata)
• Ninja Yoneda, Ninja co-Yoneda.
• Presheaf is colimit of representability (gigantic sum) = Vector can be represented as sum over basis. Representability forms a basis over cosheafs.

Yoneda lemma

• Math Stack Exchange. Naturality of bijection given by Yoneda Lemma. Several good answers.
• Peter Smith. Category Theory: A Gentle Introduction Step-by-step account of Yoneda Lemma natural isomorphism.
• Peter Smith's list of Category Theory books, notes, etc.

Is 4+1=5 like Yoneda Lemma? and with {$\rightarrow 0$} as indicating the shifts?

• {$(\alpha = a+bi, \bar{\alpha}=a-bi)\rightarrow(\alpha = a+0i,\alpha=a+0i)$}
• {$(\alpha,\beta)$} extra - is this separating, thus indicating the representation in terms of space?
• {$(\alpha,0)\rightarrow(0,0)$}

If the central term was {$(\alpha,\alpha)$}, would sameness indicate the representation in terms of time?

• Is the quadratic nature of relationships in the Yoneda lemma the basis for the {$\frac{1}{r^2}$} laws? Compare with the argument from field theory.

Yoneda lemma

• Global quantum arises from the Yoneda lemma when the finite automata reaches its limits because of the quadratic law and must be reconsidered in terms of relationships.

Yoneda Lemma

• The Yoneda Lemma interpretation in terms of A. Note that the morphism A->A is not completely symmetric. As apparent from its relation with Set. Yoneda Lemma also shows that postcomposition is different from precomposition. Thus we can introduce a functor F on the right.
• Interpret A->A as A->IA or even {$A\rightarrow I^nA$}.
• Awodey Proposition 9.9 Yoneda Lemma shows why adjoints are unique up to isomorphism. Apparently, the Yoneda Lemma allows you to remove the "Hom" in HomSets and focus on the output of the morphism. See also Proposition 9.14 that right adjoints preserve limits and left adjoints preserve colimits. And see Proposition 9.16.

Ninja Yoneda Lemma

• Jason Reed. Feeling The Yoneda Lemma in my Guts
• Fosco Loregian. Coend calculus.
• How does Goguen's hierarchy of adjunctions in systems theory (and the relation of limits and colimits) relate to the Yoneda lemma, systems and subsystems? "There is no whole system without an interconnection of its parts: and there is no whole system without its environment."

Yoneda Lemma

• Why separates input A (object) and the output (B,A) (its relations).

nlab: Geometry of Physics: Categories and Toposes

• Yoneda Lemma corresponds, in the geometry of physics, to the statement: "sets of probes of generalized spaces are indeed sets of maps from model spaces"
• Yoneda embedding statement: "nature of model spaces is preserved when regarding them as generalized spaces"; "Yoneda embedding contains and generalizes ordinary spaces"
• Yoneda embedding is free co-completion statement generalized spaces really are glued from ordinary spaces
• https://www.schoolofhaskell.com/user/bartosz/understanding-yoneda
• https://www.schoolofhaskell.com/user/bartosz/lenses-stores-and-yoneda
• Toni Mikael Annala. Stable Homotopy Without Homotopy. Applies the Yoneda Lemma at roughly 5:10.
• Compare the Yoneda lemma with the calculaton of a root space {$[e_{mm}-e_{nn},e_{ij}]=(\epsilon_i(e_{mm}-e_{nn})-\epsilon_j(e_{mm}-e_{nn}))e_{ij}$} where {$e_{mm}-e_{nn}$} is any (basis) element of the Cartan subalgebra h. Is the root {$\epsilon_i-\epsilon_j$} analogous to a natural isomorphism?
• https://mathoverflow.net/questions/431274/arrows-furnished-by-yoneda/431434#431434
• In Isbell duality, the role of Set in the relationship of presheaves and copresheaves brings to mind linear functionals, how they map to a field. And the related duality.
• Yoneda lemma in physics - relating system and subsystems - global quantum.
• https://en.m.wikipedia.org/wiki/Japanese_spider_crab
• https://www.mta.ca/~cat-dist/catlist/1999/yoneda
• Yoneda lemma - equalizer of products - are equalizers, products and initials the building blocks of all limits. Products work like sets. But you can have Yoneda lemma without sets in this way. (Riehl - Kan extensions). Is set a part of the definition of a product ? See Theorem 3.4.12
• A special case of the universal property of identity types is related to the Yoneda lemma.
• How is Yoneda lemma related to matrix row manipulation? And how might that help relate Cramer's rule to Kan extensions and the Yoneda lemma?

Riehl: 2.4. The category of elements A universal property for an object c ∈ C is expressed either by a contravariant functor F together with a representation C (−, c)  F or by a covariant functor F together with a representation C (c, −)  F. The representations define a natural characterization of the maps into (in the contravariant case) or out of (in the covariant case) the object c. Proposition 2.3.1 implies that a universal property characterizes the object c ∈ C up to isomorphism. More precisely, there is a unique isomorphism between c and any other object representing F that commutes with the chosen representations. In such contexts, the phrase “c is the universal object in C with an x” assets that x ∈ Fc is a universal element in the sense of Definition 2.3.3, i.e., x is the element of Fc that classifies the natural isomorphism that defines the representation by the Yoneda lemma. In this section, we prove that the term “universal” is being used in the precise sense alluded to at the beginning of this chapter: the universal element is either initial or terminal in an appropriate category. The category in question, called the category of elements, can be constructed in a canonical way from the data of the representable functor F. The main result of this section, Proposition 2.4.8, proves that any universal property can be understood as defining an initial or terminal object, as variance dictates, completing the promise made in §2.1.

Riehl, page 50: The Yoneda lemma is arguably the most important result in category theory, although it takes some time to explore the depths of the consequences of this simple statement. In §2.3, we define the notion of universal element that witnesses a universal property of some object in a locally small category. The universal element witnessing the universal property of the complete graph is an n-coloring of K n , an element of the set n-Color(K n ). In §2.4, we use the Yoneda lemma to show that the pair comprised of an object characterized by a universal property and its universal element defines either an initial or a terminal object in the category of elements of the functor that it represents. This gives precise meaning to the term universal: it is a synonym for either “initial” or “terminal,” with context disambiguating between the two cases. For instance, K n is the terminal n-colored graph: the terminal object in the category of n-colored graphs and graph homomorphisms that preserve the coloring of vertices.

Any functor {$F$} can be thought of as {$F:J\rightarrow C$} where {$J$} is the shape, the index set (how) and {$C$} is the image (what). And for any functor we can ask if it has a limit (an object {$L$} with maps {$\psi_X:L\rightarrow F(X)$} for all objects {$X$} of {$J$}, such that for all {$f:X\rightarrow Y$} in {$J$} we have the analogous morphisms commute in {$C$}, and that is universal as such). So that limit is Why. And is the colimit Whether? Or is whether simply the object with its identity morphism? Compare with the Yoneda lemma.

• Consider how the understanding of Yoneda lemma in terms of a left Kan extension, and in particular, the factoring, relates to the push down automata.
• Kan extensions are a framework for universality. Consider example 6.1.3, the Yoneda Lemma. Think in which ways the universality of limits, colimits, adjunctions, etc. is captured by Kan extensions. All of these universal properties can be thought of in terms of initial or terminal objects in the appropriate categories, such as the category of cones, or the comma category for the universal mapping property for adjunctions. So consider the relevant categories. How do they relate to the classification of adjoint strings?
• Mathematics is described in terms of set theory. The category of graphs {$\textrm{Set}^{A}$} where {$A$} is the category with two objects, edges E and vertices V, and two nontrivial morphisms target {$t:E\rightarrow V$} and source {$s:E\rightarrow V$}. Similarly, all mathematical structures and their structure preserving morphisms should have a similar expression in terms of sets and their relationshps. Work out various examples. Then study the role of {$\textrm{Set}^{X}$} in the Yoneda Lemma.
• Yoneda lemma lets you go from natural isomorphism of homsets to natural isomorphism of functors.
• Yoneda lemma and the shadows in Plato's cave. John Baez: "equations are shadows".

Marie La Palme Reyes, Gonzalo E. Reyes and Houman Zolfaghari. Generic figures and their glueings. A constructive approach to functor categories. Yoneda lemma.

How does the Yoneda Embedding relate to the properties of everything? Is it describing the case of an identity arrow?

• No external context -
• Accepts all things - lays down arrows
• No internal structure -
• Required concept - identity morphism

Posina Venkata Rayudu in response to Gabriel Goren Roig

• https://categorytheory.zulipchat.com/user_uploads/21317/BTTvY8p9juxp2tr3XJjAifyn/Yoneda_Lemma.pdf
• https://categorytheory.zulipchat.com/user_uploads/21317/YenSayHLO44CEonXZB7Cdjz8/Yoneda_Embedding.pdf
• https://categorytheory.zulipchat.com/user_uploads/21317/nLjLtphUxsdPStIXDpgFhRQU/Yoneda_One_Morphism.pdf

Awodey Proposition 9.9 The Yoneda Lemma is key to proving that adjoints are unique up to isomorphism.

• https://www.cantorsparadise.com/the-brilliance-of-the-yoneda-lemma-9de77c8c003f#:~:text=The%20Yoneda%20lemma%20is%20saying,category%20of%20presheaves%20on%20Spaces.
• https://mathoverflow.net/questions/3184/philosophical-meaning-of-the-yoneda-lemma
• Mike Stay (blog). The Continuation Passing Transform and the Yoneda Embedding.
• Mike Stay. The Continuation Passing Transform and the Yoneda Embedding.
• Mike Stay. The Yoneda Embedding as a Reflection.

nLab: Space and Quantity One may view the Yoneda lemma and the resulting Yoneda embedding as expressing consistency conditions on this perspective: The Yoneda lemma says that the prescribed rule for how to test a generalized space X by a test space U turns out to coincide with the actual maps from U to X, when U is itself regarded as a generalized space, and the Yoneda embedding says that, as a result, the nature of maps between test spaces does not depend on whether we regard these as test spaces or as generalized spaces.

Is there a connection between the Yoneda lemma and the Barratt–Priddy_theorem {$H_{k}(\Sigma _{n})\cong H_{k}({\text{Map}}_{0}(S^{n},S^{n}))$}?

Stuart Kauffman: You can't reduce biology to physics because biology has functions (a heart is a pump). But the Yoneda Lemma shows that mathematics can model a notion of How and distinguish it from what is not How.

• Dr.Powell. Yoneda Lemma, Cayley's theorem, etc.
• Adjoint Functors and Yoneda's Lemma

Peirce's pragmatic maxim

• "Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object."
• The Wikipedia article gives seven related versions.
• Expresses the Yoneda Lemma.
• Expresses how an algebraic object may be best understood in terms of its representations.
• Relates to significance, the conception of the nullsome, what is unencompassable. This is the level Why.
• Note that I flip pragmatism on its head. In my philosophy, it circumscribes Whether, that is, what is to be treated as real.
• https://www.reddit.com/r/math/comments/n9a761/is_it_normal_to_be_struggling_to_get_the_yoneda/