\chapter{Constructions}
\section{CPO}
Recall that $FI \iso I$ for the initial F-algebra $I$, our goal is to construct initial F-algebras as least fixpoints.
Consider for example the faculty function which is defined by the recursive equation
\[fac\;n := \mathrm{if}\; n=0\; \mathrm{then}\; 1 \;\mathrm{else}\; n \cdot fac(n-1).\]
$fac : \mathbb{N} \to \mathbb{N}$ is the \emph{solution} of the previous equation or equivalently the fixpoint of the function $\varphi : \Par(\mathbb{N},\mathbb{N}) \to \Par(\mathbb{N},\mathbb{N})$ defined by
\[\varphi\;f\;n := \mathrm{if}\; n=0\; \mathrm{then}\; 1 \;\mathrm{else}\; n \cdot f(n-1).\]
Thus, $fac$ should satisfy $fac = \varphi(fac)$, which is obtained by iteration of $\varphi$:
\begin{center}
  \begin{tabular}{l l l l l}
                                      & 0 & 1 & 2 & \ldots \\\midrule
    $\bot$                            &   &   &   &        \\
    $\varphi(\bot)$                   & 1 &   &   &        \\
    $\varphi(\varphi(\bot))$          & 1 & 1 &   &        \\
    $\varphi(\varphi(\varphi(\bot)))$ & 1 & 1 & 2 &        \\
    $\ldots$                          &   &   &   &
  \end{tabular}
\end{center}

Since $\Par(\mathbb{N},\mathbb{N})$ admits a poset structure where $f \sqsubseteq g$ iff $\dom f \subseteq \dom g$ and $\forall x \in \dom f. fx = gx$, we can view $fac$ as the supremum of the ascending chain
\[\bot \sqsubseteq \phi(\bot) \sqsubseteq \phi(\phi(\bot)) \sqsubseteq \ldots.\]

\begin{definition}[Complete Partial Order]
  A \emph{complete partial order} (\emph{CPO}) is a poset $(X, \sqsubseteq)$ with
  \begin{enumerate}
    \item the smallest element $\bot$,
    \item suprema for infinite chains:
          \[x_0 \sqsubseteq x_1 \sqsubseteq x_2 \sqsubseteq \ldots.\]
  \end{enumerate}
  We denote the supremum of such chains as $\bigsqcup_i x_i$.
\end{definition}

\begin{example} Examples of CPOs include:
  \begin{enumerate}
    \item Every set $X$ yields a CPO $(\PSet X, \subseteq)$, where the smallest element is $\emptyset$ and the supremum of chains is the big union.
    \item For any $X,Y$ we get a CPO $(\Par(X,Y), \sqsubseteq)$, defined as above. The supremum of a chain $(f_i)$ is defined by
          \[ \dom (\bigsqcup_i f_i) = \bigcup_i \dom f_i; \qquad (\bigsqcup_i f_i)(x) = f_i(x), \text{ for a suitable (big enough) } i. \]
    \item Every set $X$ yields a CPO $(X_\bot, \sqsubseteq)$, where $X_\bot = X \cup \{ \bot \}$ and $x \sqsubseteq y$ iff $x=y$ or $x = \bot$. This is called the \emph{flat CPO}.
  \end{enumerate}
\end{example}

\begin{definition}[Continuous Function]
  A function $\phi : (X, \sqsubseteq) \to (X', \sqsubseteq')$ is called \emph{continuous} if
  \begin{itemize}
    \item $\phi$ is monotonous, i.e.\ $x \sqsubseteq y \Rightarrow \phi(x) \sqsubseteq' \phi(y)$,
    \item $\phi$ preserves suprema of chains, i.e.\ $\phi(\bigsqcup x_i) = \bigsqcup' \phi(x_i)$.
  \end{itemize}
\end{definition}
\begin{example}
  The aforementioned $\phi : \Par(\mathbb{N}, \mathbb{N})$ with $\phi\;f\;n = \ite{n=0}{1}{n \cdot f(n-1)}$ is continuous.
  % TODO proof
\end{example}

\begin{theorem}[Kleene's Fixpoint Theorem]
  Let $(X, \sqsubseteq)$ be a CPO and let $\phi : X \to X$ be continuous.
  The least fixpoint of $\phi$ is
  \[\mu\phi = \bigsqcup_i\phi^i(\bot).\]
\end{theorem}
\begin{proof}
  $\mu\phi$ is a fixpoint, since
  \[\phi(\mu\phi) = \phi (\bigsqcup_i \phi^i(\bot)) = \bigsqcup_i \phi^{i+1}(\bot) = \bigsqcup_i \phi^i(\bot) = \mu\phi.\]
  To show that it is the least fixpoint, consider another fixpoint $\phi(x) = x$. To show that $\mu\phi \sqsubseteq x$ it suffices to show that $\phi^i(\bot) \sqsubseteq x$ for all $i \in \mathbb{N}$, which indeed follows by induction:
  \begin{mycase}
    \case{} $i=0$
    \[\
      \phi^0(\bot) = \bot \sqsubseteq x\]
    \case{} $i = n+1$\\
    The induction hypothesis $\phi^n \sqsubseteq x$ directly implies \[\phi^{n+1}(\bot) \sqsubseteq \phi(x) = x\] by monotonicity of $\phi$.

  \end{mycase}
\end{proof}

In the next section we will generalize this result categorically to obtain a construction for initial F-algebras.

\section{Initial Algebra Construction}
Consider a cocomplete category $\CC$, $\CC$ has similarities to a CPO, since it has
\begin{itemize}
  \item an initial object $0 \overset{\cobang_A}{\longrightarrow} A$,
  \item colimits of every $\omega$-chain of morphisms (i.e.\ diagrams $D : (\mathbb{N}, \leq) \to \CC$):
        % https://q.uiver.app/#q=WzAsNSxbMCwwLCJBXzAiXSxbMiwwLCJBXzEiXSxbNCwwLCJBXzIiXSxbNiwwLCJcXGxkb3RzIl0sWzIsMiwiQyJdLFswLDQsIlxcaW90YV8wIiwyLHsiY3VydmUiOjJ9XSxbMSw0LCJcXGlvdGFfMSJdLFsyLDQsIlxcaW90YV8yIiwwLHsiY3VydmUiOi0yfV0sWzMsNCwiXFxsZG90cyIsMCx7ImN1cnZlIjotM31dLFswLDEsImFfezAsMX0iXSxbMSwyLCJhX3sxLDJ9Il0sWzIsMywiXFxsZG90cyJdXQ==
        \[\begin{tikzcd}
            {A_0} && {A_1} && {A_2} && \ldots \\
            \\
            && C
            \arrow["{\iota_0}"', curve={height=12pt}, from=1-1, to=3-3]
            \arrow["{\iota_1}", from=1-3, to=3-3]
            \arrow["{\iota_2}", curve={height=-12pt}, from=1-5, to=3-3]
            \arrow["\ldots", curve={height=-18pt}, from=1-7, to=3-3]
            \arrow["{a_{0,1}}", from=1-1, to=1-3]
            \arrow["{a_{1,2}}", from=1-3, to=1-5]
            \arrow["\ldots", from=1-5, to=1-7]
          \end{tikzcd}\]
\end{itemize}

\begin{definition}[$\omega$-cocontinuity]
  A functor $F : \CC \to \CC$ is called \emph{$\omega$-cocontinuous} if $F$ preserves colimits of $\omega$-chains, i.e.\ if for every diagram $D: (\mathbb{N},\leq) \to \CC$:
  \[F(\oname{colim} D) \iso \oname{colim}(FD).\]
  More concretely, if for every colimit $(C, \iota)$ of $D$, the cone $(FC, F\iota)$ is the colimit of $FD$.
\end{definition}

\begin{definition}[Finitary Functor]
  A functor $F : \Set \to \Set$ is called \emph{finitary} if for every $X$ and $x \in FX$ there exists a finite surjective map $m : Y \to X$ and $y \in FY$ such that $x = F\;m\;y$.
\end{definition}

\begin{theorem}
  Finitary functors are $\omega$-cocontinuous.
\end{theorem}

\begin{lemma}\label{lem:finclosed}
  Finitary functors are closed under
  \begin{enumerate}
    \item finite products,
    \item coproducts,
    \item and composition.
  \end{enumerate}
\end{lemma}

A consequence of \autoref{lem:finclosed} is that for any signature $\Sigma$, the functor
\[F_\Sigma X = \coprod_{\sigma \in \Sigma} X^{ar(\sigma)}\]
is finitary.

\begin{theorem}
  Let $\CC$ be cocomplete and $F : \CC \to \CC$ $\sigma$-cocontinuous. Then $F$ has the initial algebra $I = \oname{colim}_{n\in\mathbb{N}} F^n 0$.
  More concretely $I$ is the colimit of the $\omega$-chain:
  % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiRjAiXSxbNCwwLCJGRjAiXSxbNiwwLCJcXGxkb3RzIl0sWzIsMiwiXFxtdSBGIl0sWzAsMSwiaSJdLFsxLDIsIkZpIl0sWzIsMywiXFxsZG90cyJdLFswLDQsIlxcaW90YV8wIiwyLHsiY3VydmUiOjJ9XSxbMSw0LCJcXGlvdGFfMSIsMl0sWzIsNCwiXFxpb3RhXzIiLDIseyJjdXJ2ZSI6LTN9XSxbMyw0LCJcXGxkb3RzIiwwLHsiY3VydmUiOi0zfV1d
  \[\begin{tikzcd}
      0 && F0 && FF0 && \ldots \\
      \\
      && {I}
      \arrow["i", from=1-1, to=1-3]
      \arrow["Fi", from=1-3, to=1-5]
      \arrow["\ldots", from=1-5, to=1-7]
      \arrow["{\iota_0}"', curve={height=12pt}, from=1-1, to=3-3]
      \arrow["{\iota_1}"', from=1-3, to=3-3]
      \arrow["{\iota_2}"', curve={height=-18pt}, from=1-5, to=3-3]
      \arrow["\ldots", curve={height=-18pt}, from=1-7, to=3-3]
    \end{tikzcd}\]
\end{theorem}

\section{Terminal Coalgebra Construction}