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See: Math notebook, Category theory, Yoneda Lemma, Learning Yoneda Lemma, Mastering the Yoneda Lemma Understand the Yoneda Lemma from a top down point of view. 自上而下的米田引理透视图。
Summary formula {$\{\{A\rightarrow\_\}\rightarrow (\_\overset{F}{\rightarrow}\overset{F(\_)}{\{\dots\}})\}\cong A\overset{F}{\rightarrow}\overset{F(A)}{\{\dots\}}$} In what sense does this mean that an element {$u\in F(A)$} is isomorphic to a natural transformation? Alternatively, we can write {$\{\{A\rightarrow\_\}\rightarrow F(\_)\}\cong F(A)$} {$\textrm{Hom}(\textrm{Hom}(A,–),F)\cong F(A)$} {$\textrm{Nat}(h^A,F)\cong F(A)$}. Key points On the left hand side, first we define A, and then F. Does that matter? Is it true on the right hand side as well? Explain each symbol in the formula. Object A in category C. Functor F from C to Set. The ambiguous notation by which a set can be interpreted as a functor. Top Down Analysis Natural isomorphism The Yoneda lemma is about a natural isomorphism {$\{\eta,\eta^{-1}\}$}. Functor category A natural isomorphism is a pair of inverse morphisms {$\eta:F\rightarrow G$} and {$\eta^{-1}:G\rightarrow F$} between functors {$F:C\rightarrow D$} and {$G:C\rightarrow D$} in the functor category {$D^C$}, where {$C$} and {$D$} are categories. Inverse morphisms {$\eta$} and {$\eta^{-1}$} are inverse morphisms in that {$\eta \eta^{-1}=1_F$} and {$\eta^{-1} \eta =1_G$}. The functor category {$U^{U^C \times C}$} In the Yoneda lemma, the relevant functor category is of the form {$U^{U^C \times C}$}. The objects are functors whose input is a functor from C to U along with an object from C, and whose output is an object from U. Thus these functors may be interpreted as taking a functor from C to U and plugging in an object from C to get an object in U. Ambiguous interpretation of {$F(A)$} Suppose we have an object {$A$} in category {$C$} and a functor {$F:C\rightarrow D$}. Then the expression {$F(A)$} may be interpreted as a functor, as above. Or it may be interpreted as an object in the codomain {$D$}. Similarly, in algebra, an expression {$f(x)$} may be intepreted ambiguously as a function of {$x$} in the range and as an output value in the codomain. The functor {$F(A)$} {$F(A)$}, as an object in {$U^{U^C \times C}$}, and thus as a functor in {$F$} and {$A$}, {$(F,A)\overset{(\psi,f)}{\rightarrow}(G,B)$}, is calculated thus: {$x \rightarrow \psi_B(F(f)(x))$}, where {$x$} is an object in {$C$}, the functor {$F$} is an object in {$U^C$}, and {$\psi_B(F(f)(x))$} is an object in {$U$}. The morphism {$(\psi , f): (F,A)\rightarrow (G,B)$} gets mapped to the set function which takes {$x \in F(A)$} to {$\psi_B(F(f)(x)) \in G(B)$}. And that set function is based on the natural transformation {$\psi$} and could also be written as {$x \mapsto G(f)(\psi_A(x))$}. Namely, the natural transformation {$F\overset{\psi}{\rightarrow}G$} gives: {$$ \matrix{ F(A) & \overset{\psi_A}{\longrightarrow} & G(A) \cr \downarrow{\scriptsize F(f)} & & \downarrow {\scriptsize G(f)} \cr F(B) & \overset{\psi_B}{\longrightarrow} & G(B) \cr } $$} See this post. So this functor plugs in {$A$} into {$F$} to yield the set {$F(A)$}. How does this functor act on a morphism {$(\alpha,f)$}? Suppose the morphism {$f$} replaces {$A$} with {$B$} and the natural transformation {$\alpha$} replaces {$F$} with {$G$}. Then how does this yield a set function that takes us from the set {$F(A)$} to the set {$G(B)$}? This is simply the set function yielded by the commutative diagram given by {$F(f)$}, {$G(f)$}, {$\alpha_A$}, {$\alpha_B$}. Note the cross relationship: {$f$} is inputted into functor {$F$} or {$G$}, and {$\alpha$} has a component for {$A$} or {$B$}. Thus "plugging in", when understood as a functor, and interpreted for a morphism {$(\alpha, f)$}, is simply the constraint on a commutativity diagram. In other words, "plugging in", understood as a functor, is in fact, the manifestation of the definition of a natural transformation from F to G. Just as for the object A it is understood as the manifestation of the morphism F. The functor category {$\mathbf{Set}^{\mathbf{Set}^C \times C}$} In the Yoneda lemma, the relevant functor category is {$\mathbf{Set}^{\mathbf{Set}^C \times C}$}. The functor category {$\mathbf{Set}^C$} First, we consider the functor category {$\mathbf{Set}^C$} whose objects are functors from {$C$} to {$\mathbf{Set}$}. We will consider two objects in {$\mathbf{Set}^C$}, a particular functor {$\textrm{Hom}(A,–):C \rightarrow \mathbf{Set}$} and an arbitrary functor {$F:C\rightarrow \mathbf{Set}$}. The particular functor {$\textrm{Hom}(A,–)$} {$\textrm{Hom}(A,X)$} denotes the set of morphisms from {$A$} to {$X$}. For any object {$A$} in {$C$}, we define a particular functor {$\textrm{Hom}(A,–)$} in {$\mathbf{Set}^C$} as follows: {$\textrm{Hom}(A,–):C \rightarrow \mathbf{Set}$} An arbitrary functor {$F:C\rightarrow \mathbf{Set}$} A functor {$F:C\rightarrow \mathbf{Set}$} is an arbitary object in {$\mathbf{Set}^C$}. It takes an object {$B$} in {$C$} to a set {$F(B)$} in {$\mathbf{Set}$}. Morphisms from {$\textrm{Hom}(A,–)$} to {$F$} The functors {$\textrm{Hom}(A,–)$} and {$F$} are objects in the functor category {$\mathbf{Set}^C$}. Thus in {$\mathbf{Set}^C$} we can consider the morphisms from {$\textrm{Hom}(A,–)$} to {$F$}. The set of morphisms from {$\textrm{Hom}(A,–)$} to {$F$} These morphisms from {$\textrm{Hom}(A,–)$} to {$F$} form the set {$\textrm{Hom}(\textrm{Hom}(A,–),F)$}. By the definition of function category, these morphisms are the natural transformations from {$\textrm{Hom}(A,–)$} to {$F$}. Thus we can alternatively write {$\textrm{Nat}(\textrm{Hom}(A,–),F)\equiv\textrm{Hom}(\textrm{Hom}(A,–),F)$}. Natural transformation What is a natural transformation? A natural transformation is a morphism {$\eta:F\rightarrow G$} in the functor category {$D^C$}. Here {$C$} and {$D$} are categories, and the functors {$F:C\rightarrow D$} and {$G:C\rightarrow D$} are objects in {$D^C$}. A natural transformation manifests the arrow {$\eta:F\rightarrow G$} as a family of morphisms {$\{\eta_X:F(X)\rightarrow G(X) | X\in ob(C)\}$}. Note that these morphisms {$\eta_X$}, called components of {$\eta$} at {$X$}, are themselves in {$D$} but are indexed by the objects {$X$} in {$C$}. Furthermore, in defining the natural transformation, for every {$f$}, these morphisms are required to satisfy the commutative diagram {$$ \matrix{ F(X) & \overset{\eta_X}{\longrightarrow} & G(X) \cr \downarrow{\scriptsize F(f)} & & \downarrow {\scriptsize G(f)} \cr F(Y) & \overset{\eta_Y}{\longrightarrow} & G(Y) \cr } $$} Note that a natural isomorphism is a special kind of natural transformation where the morphisms {$\eta_X$} are isomorphisms. This means that there is a family of inverse morphisms {$\eta^{-1}_X:G(X)\rightarrow F(X)$} such that {$\eta_X \eta^{-1}_X=1_{F(X)}$} and {$\eta^{-1}_X \eta_X =1_{G(X)}$}. The functor {$\textrm{Hom}(–,F)$} {$\textrm{Hom}(–,F)$} is an object within the functor category {$\mathbf{Set}^{\mathbf{Set}^C}$}. It takes, as input, a functor {$G:C\rightarrow \mathbf{Set}$}, and outputs the set {$\textrm{Hom}(G,F)$} of natural transformations from {$G$} to {$F$}. The set of morphisms {$\textrm{Hom}(\textrm{Hom}(A,–),F)$} {$\textrm{Hom}(\textrm{Hom}(A,–),F)$} is the set of morphisms in {$\mathbf{Set}^C$} from {$\textrm{Hom}(A,–)$} to {$F$}. These morphisms are the natural transformations from {$\textrm{Hom}(A,–)$} to {$F$}. The function {$\textrm{Hom}(\textrm{Hom}(A,–),F):C\rightarrow \mathbf{Set}$} We can interpret {$\textrm{Hom}(\textrm{Hom}(A,–),F)$} as a function which takes an object {$B$} in {$C$} and maps it to the set of morphisms {$\{\eta_B | \eta \in \textrm{Hom}(\textrm{Hom}(A,–),F)$}. Each morphism {$\eta_B$} is a set function that maps the set {$\textrm{Hom}(A,B)$} to the set {$F(B)$}. Is this function a functor from {$C$} to {$\mathbf{Set}$} ? Answer this! And can we specify the codomain more exactly and meaningfully? The functor {$\textrm{Hom}(\textrm{Hom}(A,–),F):C \times \mathbf{Set}^C\rightarrow \mathbf{Set}$} We can interpret {$\textrm{Hom}(\textrm{Hom}(A,–),F)$} as a functor that takes inputs {$A$} and {$F$} and outputs the set of morphisms {$\textrm{Hom}(\textrm{Hom}(A,–),F)$}. Eduardo: if we have two functors F,G: C -> Set and a natural transformation T: F -> G then T is essentially an operation that receives an object A of C and returns a morphism TA: FA -> GA... {$\mathbf{Set}^C \times C$} is a functor category (?) Write out the functor... (on which the natural transformation will be based) ... are there alternative ways to write out this category so that it looks indeed like a functor category? Given
The ambiguity of {$\textrm{Hom}(A,–)$} Note that the functor {$\textrm{Hom}(A,–):C\rightarrow \mathbf{Set}$} is an object in {$\mathbf{Set}^C$}. However, this expression can also be understood as a functor {$\textrm{Hom}(A,–)$} that maps input {$C$} to a different output, namely the assignment of {$C$} to the set {$\textrm{Hom}(A,C)$}. But what is the category of assignments? Readings |