概论

数学

发现

Andrius Kulikauskas

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Software


See: Foursome, Math notebook, Category theory, Adjunction, Walks, Univalence axiom

Challenge: Understand Yoneda's lemma and relate it to the four levels of knowledge: Whether, What, How, Why.



Think through and write up

  • How the properties of the Hom-functor yield Cayley's theorem.
  • Alternatively, how the Yoneda lemma yields Cayley's theorem.
  • What the Hom-functor means in the case of the Coxeter groups.

Understand

  • Universality, representability, limits, adjunctions, Kan extensions
  • Representable functors. Come up with simple examples.
  • Natural transformations. Given a natural transformation A->B, what does that say about A and B?
  • Riehl page 115: The universal property of the free vector space functor {$\mathbb{k}[−]:\textbf{Set}\rightarrow\textbf{Vect}_k$} is expressed by saying that it is left adjoint to the underlying set functor {$U:\textbf{Vect}_k\rightarrow\textbf{Set}$}: linear maps {$\mathbb{k}[S]\rightarrow\mathbf{V}$} correspond naturally to functions {$S\rightarrow U(V)$}, which specify the image of the basis vectors {$S⊂\mathbb{k}[S]$}. By the Yoneda lemma, this universal property can be used to define the action of the free vector space functor {$\mathbb{k}[−]$} on maps. The forgetful functor {$U:\textbf{Vect}_k\rightarrow\textbf{Set}$} has no right adjoint, so in this setting there are no competing notions of free construction.
  • Understand how a Kan extension expresses the generalized form of Yoneda lemma, as in Riehl's book 6.5.4, page 211.
  • Understand how the Yoneda Lemma determines a subobject classifier for a topos of sheaves.

Meaning of Yoneda lemma

  • In identifying the evaluation of a function and the execution of a step, what is the bijection saying?
  • What is meant by saying that the Yoneda Lemma expresses all of the possible relations?
  • Structuralism - elements defined by relations. How does this relate to the Yoneda lemma?
  • How can the Yoneda lemma be understood as the introduction of a parser for the subsystem?
  • How can there be a second-level parser or third-level parser?

Implicit and explicit

  • Relate "implicit" and "explicit" with sets and paths and with composition by extension and by clarification.

Perspectives

  • Compare Yoneda's lemma with perspectives, their structure, and the equation of perspectives.
  • Relate composition of perspectives with composition by extension and composition by clarification.

Information

  • Explain how substitution of eigenvalues into symmetric functions can add information. And relate this to the Yoneda lemma where associativity adds information when we go from objects (node-variables) to morphisms (matrix edges).
  • Relate functors, the idea that at best they preserve information, with the idea of nondecreasing entropy.

Actions

  • Consider the relation between things and actions (as with the natural numbers and addition) and where it shows up here, for example, in actions upon a set.

Presumptions

  • Does the Yoneda lemma presume the Axiom of Choice?
  • Does the Yoneda lemma suppose commutativity of summation? as with multiplication?

Yoneda lemma for particular categories

  • Consider the Yoneda lemma for preorders. What does it mean for Heyting algebras?
  • What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.?
  • What is the variant of Cayley's theorem that applies to the alphabet of letters i, i-bar, and yields Coxeter groups?
  • What would the category of Lists look like? And what would the Yoneda Lemma look like if the functor mapped into the category of Lists?
  • Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries?
  • Study categories with a single initial state and a single final state. What does Yoneda Lemma mean for them

Homsets

  • Consider the Yoneda Lemma as the duality of set functions and Hom Set maps. Set functions are one-directional. In what sense are Hom Set maps one-directional? And is that duality natural? And what does it mean to have a "natural" duality?

Proof

  • Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here?
  • The elements in F(A) are just dummies, they don't matter - they don't have meaning - double check, what could they possibly mean?

Identity morphism - do nothing action

  • How does the "null activity" relate to the "do nothing action"?
  • When are two identity morphisms the same?

Quantifiers

  • Analyze the Yoneda Lemma in terms of the arithmetical hierarchy. Express its statements in terms of existential and universal quantifiers.

Form of Yoneda Lemma

  • Consider how the form of Yoneda's lemma arises from the form of choices.
  • How are the Yoneda lemma (Hom sets) and (proofs by) path induction (in homotopy type theory) related?
  • Relate Yoneda Lemma to this logical tautology: {$(P \Rightarrow Q) \Rightarrow R) \Rightarrow ((P \Rightarrow Q) \Rightarrow (P \Rightarrow R))$}

Phase transition

  • Study Tai-Danae Bradley's thesis to learn about the transition between classical and quantum probabilities, between sets (F1) and vector spaces (Fq), and look for a connection with {$N$} and {$N^2$} as per the Yoneda lemma.
  • Study the symmetric functions of the eigenvalues of a random matrix.
  • Anyons are composite particles in two-dimensions that have statistics in between fermion (object) and boson (arrow) statistics. How can they be understood in terms of category theory?

Composition and Factoring out an identity morphism

  • How does the associativity of composition relate to factoring out an identity morphism?

Adjunctions

  • Relate Yoneda lemma and adjunctions.

Toposes

  • How is the Yoneda lemma related to toposes in the Applied Category Theory book?

Study Materials

Overview

Expositions

Ideas

Examples

Related Concepts

Readings

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

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.
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Šis puslapis paskutinį kartą keistas November 28, 2021, at 06:06 PM