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the Yoneda Lemma

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\BOOKMARK [1][-]{section.8}{8. Week 5}{}% 8
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\BOOKMARK [1][-]{section.10}{10. Week 6}{}% 10
\BOOKMARK [1][-]{section.11}{11. Week 7}{}% 11
\BOOKMARK [1][-]{section.11}{11. The Yoneda Lemma}{}% 11
\BOOKMARK [1][-]{section.12}{12. Week 7}{}% 12
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\subsection{} A category is called \emph{skeletal} if isomorphic objects are always identical. Show that every category is always equivalent to a skeletal one, that is to say, every category has a ``skeleton''.
\begin{comment}
\section{The Yoneda Lemma}
\iam{Nicola} In this week's talk (week 7) I skipped some details of the proof of the Yoneda lemma, so I thought I could flesh out the details here.
\iam{Nicola} In this week's talk (week 7) I skipped some details of the proof of the Yoneda lemma, so I thought I could do this here.
Let's start with some notation. Throughout I'll denote the functor category between $\mathcal{C}$ and $\mathcal{D}$ as $[\mathcal{C},\mathcal{D}]$, and the category of contravariant functors on $\mathcal{C}$ as $\mathbf{PSh}(\mathcal{C})=[\mathcal{C}^{op},\mathbf{Set}]$, which, from now on I will refer to as the category of \emph{pre-sheaves} over $\mathcal{C}$. Of particular interest is the fact that each object of $\mathcal{C}$ has a \emph{characteristic} pre-sheaf, defined by
Let's start with some notation. Throughout I'll denote the functor category between $\mathcal{C}$ and $\mathcal{D}$ as $[\mathcal{C},\mathcal{D}]$, and the category of contravariant functors on $\mathcal{C}$ as
$$\mathbf{PSh}(\mathcal{C})=[\mathcal{C}^{op},\mathbf{Set}]$$
which, from now on I will refer to as the category of \emph{pre-sheaves} over $\mathcal{C}$. Also, whenever possible without causing confusion, I will drop the dependency on the base category to denote pre-sheaves.
Of particular interest is the fact that each object of $\mathcal{C}$ has a \emph{characteristic} pre-sheaf, defined by
\begin{align*}
\yo_a:\mathcal{C}^\text{op}&\xrightarrow{\quad} \mathbf{Set}\\
b\,\, &\longmapsto \Hom_\mathcal{C}(b,a)
\end{align*}
Conversely, one also says that a given pre-sheaf $F$ is represented by an object $a$ of the source category if it comes equipped with an isomorphism of pre-sheaves between
$$F\cong \yo_a$$
This assignment, to each object its characteristic pre-sheaf, is natural as it arises from the hom-functor trough the product/functor-category adjunction in $\mathbf{Cat}$. Consequently, it arranges itself into a functor called the Yoneda embedding
\begin{align*}
\yo:\mathcal{C}&\xrightarrow{\quad} \mathbf{PSh}(\mathcal{C})\\
a\,\,&\longmapsto \,\yo_a
\end{align*}
We are now ready to state the main lemma.
\begin{lem}(Yoneda) Let $\mathcal{C}$ be a locally small category. For every pre-sheaf $G\in[\mathcal{C}^{op},\mathbf{Set}]$ and object $a\in \mathcal{C}_0$ there is a canonical isomorphism
\begin{align*}
\mathrm{Nat}(\yo_a,G)\cong G(a)
\Phi:\mathrm{Nat}(\yo_a,G)\cong G(a)
\end{align*}
between the object of natural transformations, from the characteristic pre-sheaf of $a$ to $G$, and the value of $G$ at $a$. Moreover, this isomorphism is natural both in $a$ and $F$.
between the object of natural transformations, from the characteristic pre-sheaf of $a$ to $G$, and the value of $G$ at $a$. Moreover, this isomorphism is natural both in $a$ and $G$.
\end{lem}
\end{comment}
This is the most precise version of the Yoneda lemma, and, without a doubt, the one everyone should know. However, I would like to rephrase it in a weaker, but, I believe, more suggestive way.
The reason that we can do this is that $\mathbf{Cat}$, the category of \emph{small} categories and functors (recall that a small category has a \emph{set} of objects), is cartesian closed with its exponentials given by
\begin{align*}
\mathcal{C}^\mathcal{D}=[\mathcal{D},\mathcal{C}]
\end{align*}
So that we obtain an evaluation functor, which in the particular case of pre-sheaves looks like
\begin{align*}
ev:\mathcal{C}^{op}\times \mathbf{PSh}\longrightarrow \mathbf{Set}\\
(a,F)\longmapsto F(a)
\end{align*}
At the same time, the assignment to each object its characteristic pre-sheaf, is natural as it arises from the hom-functor trough the product/functor-category adjunction in $\mathbf{Cat}$ (a topic which we will encounter in two weeks). Consequently, it arranges itself into a functor called the Yoneda embedding
\begin{align*}
\yo:\mathcal{C}&\xrightarrow{\quad} \mathbf{PSh}\\
a\,\,&\longmapsto \,\yo_a
\end{align*}
Suspending for a moment your disbelief, consider the following composition
\begin{center}
\begin{tikzcd}
\mathcal{C}^{op}\times \mathbf{PSh} \arrow[rr, "\yo^{op}\times \mathrm{id}"] & & \mathbf{PSh}^{op}\times \mathbf{PSh} \arrow[rr, "\Hom_{\mathbf{PSh}}"] & & \mathbf{Set}
\end{tikzcd}
\end{center}
It acts on objects by mapping $a\in \mathcal{C}_0$ and $G:\mathcal{C}^{op}\to \mathbf{Set}$ to
\begin{align*}
(a,G)\longmapsto (\yo_a,G)\longmapsto \mathrm{Nat}(\yo_a,G)
\end{align*}
where the last morphism evaluates to natural transformations because in the functor category $\mathbf{PSh}(\mathcal{C})$ morphisms are given by $\Hom_{\mathbf{PSh}}(F,G) = \mathrm{Nat}(F,G)$.
Both this composition and the evaluation functor have source $\mathcal{C}^{op}\times \mathbf{PSh}$ and values in $\mathbf{Set}$. It is not hard to see how, together, the isomorphisms of the Yoneda lemma and the statement of their naturality arrange themselves into a natural isomorphism connecting these two functors, at least in the special case of \emph{small} categories where these are defined.
\begin{center}
\begin{tikzcd}
& \; \arrow[dd, Rightarrow, "\Phi"',shorten >=1.5ex, shorten <=1.5ex] & \\
\mathcal{C}^{op}\times \mathbf{PSh} \arrow[rr, "{\mathrm{Nat}(\yo_{(-)},-)}", bend left=49] \arrow[rr, "ev"', bend right=49] & & \mathbf{Set} \\
& \; &
\end{tikzcd}
\end{center}
Our immediate goal is to do this: defining $\Phi$, proving that it is natural, and proving that it is an iso.
Recall that natural transformations are defined \emph{point-wise} on the source, meaning that to define $\Phi$ we need to specify a function (in $\mathbf{Set}$) for each $a\in \mathcal{C}_0$ and $F\in \mathbf{PSh}$. This function is defined as follows
\begin{align*}
\Phi_{a,F}:\mathrm{Nat}(\yo_a,F)&\longrightarrow F(a)\\
(\alpha:\yo_a \xLongrightarrow{} F)&\mapsto \alpha_a(\id_a)
\end{align*}
by the observation that, amongst more data, a natural transformation $\alpha:\yo_a\Rightarrow F$ specifies a function
\begin{align*}
\alpha_a: \yo_a(a)=\Hom_\mathcal{C}(a,a)&\longrightarrow F(a)\\
\id_a &\longmapsto \alpha_a(\id_a)
\end{align*}
which justifies how $\alpha_a(\id_a)\in F(a)$.
Firstly, we will prove that this family of functions is natural, hence that $\Phi$ is indeed a natural transformation.
\begin{proof}[Proof of Naturality] In order to prove the naturality of $\Phi$ we will need to check that two diagrams commute, one that has to do with morphisms in $\mathcal{C}$, and one that takes care of natural transformations in $\mathbf{PSh}$. And, while perhaps an expert in category theory could conjure these with ease, let us take a 2-categorical point of view to understand where these come from.
As we mentioned now multiple times, a functor from the terminal category $\star$ to any generic category $\mathcal{D}$ acts by selecting a single object, which by convention we take to label the functor. Similarly, a morphism $f:a\to b$ in $\mathcal{D}$ can be expressed as a natural transformation
\begin{center}
\begin{tikzcd}
& \; \arrow[dd, Rightarrow, "f"',shorten >=1.5ex, shorten <=1.5ex]& \\
\star \arrow[rr, "a", bend left=49] \arrow[rr, "b"', bend right=49] & & \mathcal{D} \\
& \; &
\end{tikzcd}
\end{center}
I repeat this because it gives what I believe to be a nice way to visualize the componenets of $\Phi$. Specifically, if one considers the diagram below, there are two functors from the terminal category to $\mathbf{Set}$, each of which selects a set, and the natural transformation between them which ``selects'' the component $\Phi_{a,F}$ of $\Phi$.
\begin{center}
\begin{tikzcd}
& & & \; \arrow[dd, Rightarrow, "\Phi"',shorten >=1.5ex, shorten <=1.5ex]& \\
\star \arrow[rr, "{\{a,F\}}"] & & \mathcal{C}^{op}\times \mathbf{PSh} \arrow[rr, "{\mathrm{Nat}(\yo_{(-)},-)}", bend left=49] \arrow[rr, "ev"', bend right=49] & & \mathbf{Set} \\
& & & \; &
\end{tikzcd}
\end{center}
Now let us fix $F$ (one could in principle do both $a$ and $F$ at the same time but this is not very wise), and consider a morphism $f:a\to b$ in $\mathcal{C}$
\begin{center}
\begin{tikzcd}
& \; \arrow[dd, Rightarrow, "f^{op}\times \id",shorten >=1.5ex, shorten <=1.5ex] & & \; \arrow[dd, Rightarrow, "\Phi"',shorten >=1.5ex, shorten <=1.5ex]& \\
\star \arrow[rr, "{\{a,F\}}"', bend right=49] \arrow[rr, "{\{b,F\}}", bend left=49] & & \mathcal{C}^{op}\times \mathbf{PSh} \arrow[rr, "{\mathrm{Nat}(\yo_{(-)},-)}", bend left=49] \arrow[rr, "ev"', bend right=49] & & \mathbf{Set} \\
& \; & & \; &
\end{tikzcd}
\end{center}
Now there are four different paths going from the terminal category to $\mathbf{Set}$, connected each by natural transformations. If we evaluate (as in look at their images) in $\mathbf{Set}$ these yield a square
\begin{center}
\begin{tikzcd}
{\mathrm{Nat}(\yo_a,F)} \arrow[dd, "{\Phi_{a,F}}"] & & {\mathrm{Nat}(\yo_b,F)} \arrow[ll, "\yo(f)_*"'] \arrow[dd, "{\Phi_{b,F}}"] \\
& & \\
F(a) & & F(b) \arrow[ll, "F(f)"]
\end{tikzcd}
\end{center}
whose commutativity guarantee naturality in $a\in \mathcal{C}_0$.
\emph{Try to convince yourself of this fact, i.e. that naturality means that whichever path you take you should commute in the target}.
To see that this square commutes, pick a natural transformation $\beta\in \mathrm{Nat}(\yo_b,F)$ and run trough the diagram
\begin{center}
\begin{tikzcd}
(\yo_a \xRightarrow{\yo(f)} \yo_b\xRightarrow{\beta} F) \arrow[dd, "{\Phi_{a,F}}", maps to] & & (\yo_b\xRightarrow{\beta} F) \arrow[ll, "\yo(f)_*"', maps to] \arrow[dd, "{\Phi_{b,F}}", maps to] \\
& & \\
{} \,\, & & \beta_b(\id_b) \in F(b) \arrow[ll, "F(f)", maps to]
\end{tikzcd}
\end{center}
This gives an equation
\begin{align*}
F(f)(\beta_b(\id_b))=& (\beta\circ \yo(f))_a(\id_a)\\
=& \beta_a(f)
\end{align*}
which is satisfied because the naturality of $\beta$ implies the commutativity of the following diagram
\begin{center}
\begin{tikzcd}
\yo_b(b) \arrow[dd, "\yo_b(f)"] \arrow[rr, "\beta_b"] & & F(b) \arrow[dd, "F(f)"] & & \id_b \arrow[dd, maps to] \arrow[rr, maps to] & & \beta_b(\id_b) \arrow[dd, maps to] \\
& & & & & & \\
\yo_b(a) \arrow[rr, "\beta_a"] & & F(a) & & f \arrow[rr, maps to] & & \beta_a(f)=F(f)(\beta_b(\id_b))
\end{tikzcd}
\end{center}
This proves naturality with respect to $a\in \mathcal{C_0}$. In a similar fashion, to prove naturality in $F\in \mathbf{PSh}$, fix an object $a\in \mathcal{C}$ and pick a natural transformation $(\gamma:F\Rightarrow G)\in \mathbf{PSh}$. We again express it as
\begin{center}
\begin{tikzcd}
& \; \arrow[dd, Rightarrow, " \id\times \gamma",shorten >=1.5ex, shorten <=1.5ex]& & \; \arrow[dd, Rightarrow, " \Phi"',shorten >=1.5ex, shorten <=1.5ex] & \\
\star \arrow[rr, "{\{a,G\}}"', bend right=49] \arrow[rr, "{\{a,F\}}", bend left=49] & & \mathcal{C}^{op}\times \mathbf{PSh} \arrow[rr, "{\mathrm{Nat}(\yo_{(-)},-)}", bend left=49] \arrow[rr, "ev"', bend right=49] & & \mathbf{Set} \\
& \; & & \; &
\end{tikzcd}
\end{center}
and look at the image that the terminal category has in $\mathbf{Set}$ with respect to these functors and natural transformations. They yield the square which commutes because, choosing a natural transformation $\alpha \in \mathrm{Nat}(\yo_a, F)$, one obtains
\begin{center}
\begin{tikzcd}
{\mathrm{Nat}(\yo_a,F)} \arrow[rr, "\gamma_*"] \arrow[dd, "{\Phi_{a,F}}"] & & {\mathrm{Nat}(\yo_a,G)} \arrow[dd, "{\Phi_{a,G}}"] & & (\alpha:\yo_a\Rightarrow F)) \arrow[rr, maps to] \arrow[dd, maps to] & & (\yo_a \xRightarrow{\alpha}F\xRightarrow{\gamma} G) \arrow[dd, maps to] \\
& & & & & & \\
F(a) \arrow[rr, "\gamma_a"] & & G(a) & & \alpha_a(\id_a) \arrow[rr, maps to] & & (\gamma\circ \alpha)_a(\id_a)
\end{tikzcd}
\end{center}
This finishes the proof that $\Phi$ is a well defined natural transformation.
\end{proof}
Now that we know that $\Phi$ is well defined, the Yoneda lemma for small categories reduces to the statement that $\Phi$ is a natural isomorphism, which we prove now.
\begin{proof}[Proof of Isomorphism] As stated in week 6, a natural transformation is a natural isomorphism if and only if it is an isomorphism \emph{point-wise}. Hence, it suffices for us to show that, having fixed $a\in \mathcal{C}_0$ and $F:\mathcal{C}^{op}\to \mathbf{Set}$, the function
\begin{align*}
\Phi_{a,F}:\mathrm{Nat}(\yo_a,F)&\longrightarrow F(a)\\
(\alpha:\yo_a \xLongrightarrow{} F)&\mapsto \alpha_a(\id_a)
\end{align*}
is a bijection. We do this, of course, constructing its inverse.
Hence, let us define a function that assigns to each point $x\in F(a)$ a natural transformation
\begin{align*}
\Gamma_{a,F}:F(a)&\longrightarrow \mathrm{Nat}(\yo_a,F)\\
x&\longmapsto (\chi :\yo_a\Rightarrow F)
\end{align*}
defined, as usually, point-wise for each $c\in \mathcal{C}$
\begin{align*}
\chi_c: (\yo_a(b)=\Hom_\mathcal{C}(c,a))&\longrightarrow F(c)\\
(g:c\to a)&\longmapsto F(g)(x)\in F(c)
\end{align*}
which is well defined because, in the image of $F$, we have
\begin{align*}
F(a) &\xrightarrow{F(g)} F(c)\\
x&\longmapsto F(g)(x)
\end{align*}
Now, to show that these two functions are inverse, pick $x\in F(a)$ and build the natural transformation $\chi:\yo_a\to F$ as above. Then, to see what this natural transformation is mapped to under $\Phi_{a,F}$ we need to evaluate it at $a$
\begin{align*}
\chi_a:\yo_a(a)&\longrightarrow F(a)\\
\id_a &\longmapsto F(\id_a)(x)= \id_{F(a)}(x)=x
\end{align*}
where we used the functoriality of $F$. This proves that $\Gamma$ is a right inverse to $\Phi$.
In the other direction, let us start with a natural transformation $\alpha:\yo_a\Rightarrow F$. Under $\Phi$ this is mapped to $\alpha_a(\id_a)\in F(a)$, as we have already seen. Pushing forward, this element of $F(a)$ yields under $\Gamma_{a,F}$ a natural transformation $\chi:\yo_a\Rightarrow F$ which, at $c\in \mathcal{C}_0$, evaluates to
\begin{align*}
\chi_c: (\yo_a(b)=\Hom_\mathcal{C}(c,a))&\longrightarrow F(c)\\
(g:c\to a)&\longmapsto F(g)(\alpha_a(\id_a))
\end{align*}
Then, $\Phi$ and $\Gamma$ are inverses if $\chi_c(g)=\alpha_c(g)=F(g)(\alpha_a(\id_a))$, which we see by observing that the naturality of $\alpha$ implies that for all $g:c\to a$ the following diagram commutes
\begin{center}
\begin{tikzcd}
\yo_a(a) \arrow[dd, "\yo_a(g)"] \arrow[rr, "\alpha_a"] & & F(a) \arrow[dd, "F(g)"] & & \id_a \arrow[dd, maps to] \arrow[rr, maps to] & & \alpha_a(\id_a) \arrow[dd, maps to] \\
& & & & & & \\
\yo_a(c) \arrow[rr, "\alpha_c"] & & F(c) & & g \arrow[rr, maps to] & & \alpha_c(g)=F(g)(\alpha_a(\id_a))
\end{tikzcd}
\end{center}
We now know that $\Phi_{a,F}$ has an inverse $\Gamma_{a,F}$ for all choices of $a$ and $F$, hence it is an isomorphism in all components and consequently a natural isomorphism. This concludes the proof.
\end{proof}
Before signing off let me comment on why the Yoneda lemma is not usually stated this way, as a natural transformation being a natural isomorphism. The problem is, as all ugly things are in category theory, to do with set-theoretic issues. Specifically, all the categories that we consider are \emph{locally small}, which I remind you means that between any two objects is a \emph{set} of morphism. However, as we very conveniently avoided to say, functor categories between locally small categories are not usually locally small, meaning that between any two functors there could be a \emph{class} of natural transformation instead of a set. It is not hard to see why this is, a single natural transformation is indexed by the objects in the source, and, if this is a class of objects (rather then a set), then it is only by a miracle that natural transformation could form a set.
Remarkably, as a side remark, this happens as a consequence of the Yoneda lemma. The bijection
\begin{align*}
\Phi:\mathrm{Nat}(\yo_a,G)\cong G(a)
\end{align*}
implies that $\mathrm{Nat}(\yo_a,G)$ is a set!
Now, what breaks if we consider locally small categories? First of all we need to be incredibly careful on how we even define functor categories between them, or even pre-sheaves categories over them. Specifically, we would need to define a new category $\mathbf{SET}$ of large sets (whatever this means), and have our categories be enriched over this (we will see what enrichment means later). Then one obtains $\mathbf{CAT}$, the category of large category, which, for all intents and purposes, should be cartesian closed and possesses evaluation functors. However, saying precisely what this are is rather complicated. And, while one \emph{can} write a diagram like
\begin{center}
\begin{tikzcd}
& \; \arrow[dd, Rightarrow, "\Phi"',shorten >=1.5ex, shorten <=1.5ex] & \\
\mathcal{C}^{op}\times \mathbf{PSh} \arrow[rr, "{\mathrm{Nat}(\yo_{(-)},-)}", bend left=49] \arrow[rr, "ev"', bend right=49] & & \mathbf{SET} \\
& \; &
\end{tikzcd}
\end{center}
I'm sure that this would turn any logician purple.
\section{Week 7}
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\contentsline {subsection}{\tocsubsection {}{10.8}{}}{30}{subsection.10.8}
\contentsline {subsection}{\tocsubsection {}{10.9}{}}{30}{subsection.10.9}
\contentsline {section}{\tocsection {}{11}{Week 7}}{30}{section.11}
\contentsline {subsection}{\tocsubsection {}{11.1}{}}{30}{subsection.11.1}
\contentsline {subsection}{\tocsubsection {}{11.2}{}}{30}{subsection.11.2}
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\contentsline {subsection}{\tocsubsection {}{11.5}{}}{30}{subsection.11.5}
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\contentsline {section}{\tocsection {}{11}{The Yoneda Lemma}}{30}{section.11}
\contentsline {section}{\tocsection {}{12}{Week 7}}{35}{section.12}
\contentsline {subsection}{\tocsubsection {}{12.1}{}}{35}{subsection.12.1}
\contentsline {subsection}{\tocsubsection {}{12.2}{}}{35}{subsection.12.2}
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\contentsline {subsection}{\tocsubsection {}{12.7}{}}{35}{subsection.12.7}
\contentsline {subsection}{\tocsubsection {}{12.8}{}}{36}{subsection.12.8}
\contentsline {subsection}{\tocsubsection {}{12.9}{}}{36}{subsection.12.9}
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