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
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.
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.
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
\[F_\Sigma X =\coprod_{\sigma\in\Sigma} X^{ar(\sigma)}\]
