Finish functor section

tex/.vscode/ltex.dictionary.en-US.txt

@@ -36,3 +36,4 @@ hom-functor
 monoids
 n-ary
 Cocartesian
+Yoneda

tex/sections/02_categories.tex

@@ -379,6 +379,122 @@ Isomorphisms of categories are the isomorphisms in the quasi-category $\emph{CAT
 \end{example}
 
 \section{Natural Transformations}
+Natural transformation are morphisms between functors. The definition of ``naturality'' was one of the original goals of category theory.
+\begin{definition}[Natural Transformation]
+  Given two functors $F, G : \C \rightarrow \D$.
+  A natural transformation $\alpha : F \rightarrow G$ between these functors is a family of morphisms $(\alpha_C : FC \rightarrow GC)_{C\in\obj{\C}}$, such that for any $f : A \rightarrow B$ the diagram
+  % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGQSJdLFsyLDAsIkZCIl0sWzAsMiwiR0EiXSxbMiwyLCJHQiJdLFswLDEsIkZmIl0sWzIsMywiR2YiXSxbMCwyLCJcXGFscGhhX0EiLDJdLFsxLDMsIlxcYWxwaGFfQiJdXQ==
+  \[
+    \begin{tikzcd}
+      FA && FB \\
+      \\
+      GA && GB
+      \arrow["Ff", from=1-1, to=1-3]
+      \arrow["Gf", from=3-1, to=3-3]
+      \arrow["{\alpha_A}"', from=1-1, to=3-1]
+      \arrow["{\alpha_B}", from=1-3, to=3-3]
+    \end{tikzcd}
+  \]
+  commutes.
+\end{definition}
+
+\begin{example} Examples of natural transformations include:
+  \begin{enumerate}
+    \item The obvious function $flatten : Tree\;A \rightarrow List\;A$:
+          % https://q.uiver.app/#q=WzAsNCxbMCwwLCJUcmVlXFw7QSJdLFsyLDAsIlRyZWVcXDtCIl0sWzAsMiwiTGlzdFxcO0EiXSxbMiwyLCJMaXN0XFw7QiJdLFswLDIsImZsYXR0ZW5fQSIsMl0sWzEsMywiZmxhdHRlbl9CIl0sWzIsMywibGlzdFxcO2YiXSxbMCwxLCJ0cmVlXFw7ZiJdXQ==
+          \[
+            \begin{tikzcd}
+              {Tree\;A} && {Tree\;B} \\
+              \\
+              {List\;A} && {List\;B}
+              \arrow["{flatten_A}"', from=1-1, to=3-1]
+              \arrow["{flatten_B}", from=1-3, to=3-3]
+              \arrow["{list\;f}", from=3-1, to=3-3]
+              \arrow["{tree\;f}", from=1-1, to=1-3]
+            \end{tikzcd}
+          \]
+    \item For $Id, Q : Set \rightarrow Set$ we have $\delta : Id \rightarrow Q$ given by $\delta_X (x) = (x,x)$.
+    \item On $\mathcal{P}$ we can define natural transformations $\eta : Id \rightarrow \mathcal{P}$ and $\mu : \mathcal{P}\mathcal{P} \rightarrow \mathcal{P}$ by:
+          \begin{alignat*}{1}
+            \eta_X : X & \rightarrow \mathcal{P}X \\
+            x          & \mapsto \{x\}
+          \end{alignat*}
+          and
+          \begin{alignat*}{1}
+            \mu_X : \mathcal{P}\mathcal{P}X & \rightarrow \mathcal{P}X \\
+            Z                               & \mapsto \bigcup Z.
+          \end{alignat*}
+    \item Between $Q$ and $\mathcal{P}$ we can consider $\alpha,\beta : Q \rightarrow \mathcal{P}$ given by
+          \begin{alignat*}{2}
+             & \alpha_X(x,y) &  & = \{x,y\} \\
+             & \beta_X(x,y)  &  & = \{x\}.
+          \end{alignat*}
+  \end{enumerate}
+\end{example}
+
+Functors $\C \rightarrow \D$ together with natural transformations as morphisms form a quasi-category $[\C,\D]$, that is called the functor category. If $\C$ is small, then $[\C,\D]$ is a category, where identity and composition are defined component wise.
+
+\begin{example} Let us examine concrete examples of functor categories:
+  \begin{enumerate}
+    \item $[2, \C] \cong \C \times \C$, where $2$ is the \emph{discrete} category with two objects, i.e.\ $2$ has no morphisms besides the identities.
+    \item Let $\rightarrow$ be the category with 2 objects and a single non-trivial morphism $m$. $[\rightarrow, \C$ is the \emph{category of morphisms} of $\C$, where morphisms $Fm \rightarrow Gm$ are pairs of morphisms $(f,g)$ where
+          % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGMCJdLFsyLDAsIkYxIl0sWzAsMiwiRzAiXSxbMiwyLCJHMSJdLFswLDEsIkZtIl0sWzIsMywiR20iXSxbMCwyLCJmIiwyXSxbMSwzLCJnIl1d
+          \[
+            \begin{tikzcd}
+              F0 && F1 \\
+              \\
+              G0 && G1
+              \arrow["Fm", from=1-1, to=1-3]
+              \arrow["Gm", from=3-1, to=3-3]
+              \arrow["f"', from=1-1, to=3-1]
+              \arrow["g", from=1-3, to=3-3]
+            \end{tikzcd}
+          \]
+          commutes.
+  \end{enumerate}
+\end{example}
+
+\begin{definition}[Natural Isomorphism]
+  Isomorphisms in $[\C,\D]$ are called \emph{natural isomorphisms}.
+\end{definition}
+
+\begin{proposition}
+  $\alpha : F \rightarrow G$ is a natural isomorphism \emph{iff} every $\alpha_C$ is an isomorphism.
+\end{proposition}
+
+\begin{example} Let us consider some examples of natural isomorphisms:
+  \begin{enumerate}
+    \item In $[\emph{Set},\emph{Set}]$ is $Id \cong \emph{Set}(1,-)$, since of course $Id\;X = X \cong X^1 = \emph{Set}(1,X)$.
+    \item Also in $[\emph{Set},\emph{Set}]$ is $Q \cong \emph{Set}(2,-)$, similarly is $\lambda X.2\times X \cong \lambda X. X + X$.
+    \item The forgetful functor $U : \emph{Pos} \rightarrow \emph{Set}$ is naturally isomorphic to $\emph{Pos}(1,-)$, because the constant mapping $x : 1 \rightarrow X$ is monotonous for every element $x$ of a poset.
+  \end{enumerate}
+\end{example}
+
+\begin{proposition}[Yoneda Lemma]
+  Let $A \in \obj{\C}$ and $G : \C \rightarrow \emph{Set}$. Then the natural transformations
+  \[\C(A,-) \rightarrow G\]
+  are in bijection with the elements of the set $GA$.
+\end{proposition}
+\begin{proof}
+  The mappings are
+  \begin{alignat*}{1}
+     & Z : GA  \rightarrow [\C , \emph{Set}](\C(A,-), G) \\
+     & Z\;x\;h = G\;h\;x
+  \end{alignat*}
+  and
+  \begin{alignat*}{1}
+     & Y : [\C , \emph{Set}](\C(A,-), G) \rightarrow GA   \\
+     & Y\;\alpha                         = \alpha_A\;id_A
+  \end{alignat*}
+\end{proof}
+
+\begin{example}
+  Let us consider an application of the Yoneda Lemma: how many natural transformations $Id \rightarrow Q$ are there?
+  Recall that $Id \cong \emph{Set}(1,-)$, and by Yoneda there is exactly $\vert Q1 \vert = 1$ natural transformation $\emph{Set}(1,-) \rightarrow Q$, thus the number of natural transformations $Id \rightarrow Q$ is $1$.
+
+  Furthermore, consider the number of natural transformations $Q \rightarrow Q$. Recall that $Q \cong \emph{Set}(2, -)$, and by Yoneda there are $\vert Q2 \vert = 4$ natural transformations $\emph{Set}(2, -) \rightarrow Q$, thus the number of natural transformations $Q \rightarrow Q$ is $4$.
+\end{example}
+
 \section{Functor Algebras}
 \section{Functor Coalgebras}
 \section{(co)Limits} % chktex 36