\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{initial algebra of an endofunctor}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{induction}{}\paragraph*{{Induction}}\label{induction}

[[!include induction - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_algebras_over_a_monad}{Relation to algebras over a monad}\dotfill \pageref*{relation_to_algebras_over_a_monad} \linebreak
\noindent\hyperlink{LambeksTheorem}{Lambek's theorem}\dotfill \pageref*{LambeksTheorem} \linebreak
\noindent\hyperlink{AdameksTheorem}{Ad\'a{}mek's theorem}\dotfill \pageref*{AdameksTheorem} \linebreak
\noindent\hyperlink{semantics_for_inductive_types}{Semantics for inductive types}\dotfill \pageref*{semantics_for_inductive_types} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{NaturalNumbers}{Natural numbers}\dotfill \pageref*{NaturalNumbers} \linebreak
\noindent\hyperlink{more_examples}{More examples}\dotfill \pageref*{more_examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An \textbf{initial [[algebra for an endofunctor|algebra]]} for an [[endofunctor]] $F$ on a [[category]] $C$ is an [[initial object]] in the category of [[algebra over an endofunctor|algebras]] of $F$. These play a role in particular as the [[categorical semantics]] for [[inductive types]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_algebras_over_a_monad}{}\subsubsection*{{Relation to algebras over a monad}}\label{relation_to_algebras_over_a_monad}

The concept of an [[algebra of an endofunctor]] is arguably somewhat odd, a more natural concept being that of an [[algebra over a monad]]. However, the former can often be reduced to the latter.

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is a [[complete category]], then the [[category]] of [[algebras of an endofunctor]] $F : \mathcal{C} \to \mathcal{C}$ is [[equivalence of categories|equivalent]] to the category of [[algebras over a monad]] of the [[free monad]] on $F$, if the latter exists.

\end{prop}
The proof is fairly straightforward, see for instance (\hyperlink{Maciej}{Maciej}) or at [[free monad]].

The existence of free monads, on the other hand, can be a tricky question. One general technique is the [[transfinite construction of free algebras]].

\hypertarget{LambeksTheorem}{}\subsubsection*{{Lambek's theorem}}\label{LambeksTheorem}

\begin{theorem}
\label{}\hypertarget{}{}
If $F$ has an initial algebra $\alpha: F(X) \to X$, then $X$ is [[isomorphism|isomorphic]] to $F(X)$ via $\alpha$.

\end{theorem}
\begin{remark}
\label{}\hypertarget{}{}
In this sense, $X$ is a fixed point of $F$. Being initial, $X$ is the \emph{smallest} fixed point of $F$ in that there is a map from $X$ to any other fixed point (indeed, any other algebra), and this map is an [[injection]] if $C$ is [[Set]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The dual concept is [[terminal coalgebra]], which is the \emph{largest} fixed point of $F$.

\end{remark}
\begin{proof}
Given an initial algebra structure $\alpha: F(X) \to X$, define an algebra structure on $F(X)$ to be $F(\alpha): F(F(X)) \to F(X)$. By initiality, there exists an $F$-algebra map $i: X \to F(X)$, so that

\begin{displaymath}
\itexarray{
F(X) & \overset{F(i)}{\to} & F(F(X)) \\
\alpha \downarrow & & \downarrow F(\alpha) \\
X & \underset{i}{\to} & F(X)
}
\end{displaymath}
commutes. Now it is trivial, in fact tautological that $\alpha$ is itself an $F$-algebra map $F(X) \to X$. Thus $\alpha \circ i = 1_X$, since both sides of the equation are $F$-algebra maps $X \to X$ and $X$ is initial. As a result, $F(\alpha) \circ F(i) = 1_{F(X)}$, so that $i \circ \alpha = 1_{F(X)}$ according to the commutative square. Hence $\alpha$ is an isomorphism, with inverse $i$.

\end{proof}
\hypertarget{AdameksTheorem}{}\subsubsection*{{Ad\'a{}mek's theorem}}\label{AdameksTheorem}

In many cases, initial algebras can be constructed in [[recursion|recursive]] fashion, using the following special case of a theorem due to Ad\'a{}mek.

\begin{theorem}
\label{AdameksTheorem}\hypertarget{AdameksTheorem}{}
Let $C$ be a category with an [[initial object]] $0$ and [[transfinite composition]] of length $\omega$, hence [[colimits]] of sequences $\omega \to C$ (where $\omega$ is the first infinite [[ordinal]]), and suppose $F: C \to C$ preserves colimits of $\omega$-chains. Then the colimit $\gamma$ of the chain

\begin{displaymath}
0 \overset{i}{\to} F(0) \overset{F(i)}{\to} \ldots \to F^{(n)}(0) \overset{F^{(n)}(i)}{\to} F^{(n+1)}(0) \to \ldots
\end{displaymath}
carries a structure of initial $F$-algebra.

\end{theorem}
\begin{proof}
The $F$-algebra structure $F\gamma \to \gamma$ is inverse to the canonical map $\gamma \to F\gamma$ out of the colimit (which is invertible by the hypothesis on $F$). The proof of initiality may be extracted by dualizing the corresponding proof given at [[terminal coalgebra]].

\end{proof}
This approach can be generalized to the [[transfinite construction of free algebras]].

\hypertarget{semantics_for_inductive_types}{}\subsubsection*{{Semantics for inductive types}}\label{semantics_for_inductive_types}

Initial algebras of endofunctors are the [[categorical semantics]] of [[extensional type theory|extensional]] [[inductive types]]. The generalization to [[weak initial]] algebras captures the notion in [[intensional type theory]] and [[homotopy type theory]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{NaturalNumbers}{}\subsubsection*{{Natural numbers}}\label{NaturalNumbers}

The archetypical example of an initial algebra is the set of [[natural numbers]].

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{T}$ be [[topos]] and let $F : \mathcal{T} \to \mathcal{T}$ the functor given by

\begin{displaymath}
F : X \mapsto * \coprod X
\end{displaymath}
(hence the functor underlying the ``[[maybe monad]]''). Then an initial algebra over $F$ is precisely a [[natural number object]] $\mathbb{N}$ in $\mathcal{T}$.

\end{prop}
\begin{proof}
By definition, an $F$-algebra is an object $X$ equipped with a morphism

\begin{displaymath}
(0,s) :  * \coprod X \to X
  \,,
\end{displaymath}
hence equivalently with a [[pointed object|point]] $0 : * \to X$ and an [[endomorphism]] $s : X \to X$. This being inital means that for $(0_Y, s_Y) : * \coprod Y \to Y$ any other morphism, there is a unique morphism $f : X \to Y$ such that the [[diagram]]

\begin{displaymath}
\itexarray{
    * &\stackrel{0}{\to}& X &\stackrel{s}{\to}& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}}
    \\
    * &\stackrel{0}{\to}& Y &\stackrel{s}{\to}& Y  
  }
\end{displaymath}
commutes. This is the very definition of [[natural number object]] $X = \mathbb{N}$.

\end{proof}
\hypertarget{more_examples}{}\subsubsection*{{More examples}}\label{more_examples}

Theorem \ref{AdameksTheorem} applies in particular to any functor $F: Set \to Set$ which is a [[colimit]] of finitely [[representable functors]] $hom(n, -): X \mapsto X^n$, as in the following examples.

\begin{itemize}%
\item Let $A$ be a set, and let $F: Set \to Set$ be the functor $F(X) = 1 + A \times X$. Then the initial $F$-algebra is $A^*$, the [[free monoid]] on $A$. The $F$-algebra structure is

\begin{displaymath}
(e, m| ): 1 + A \times A^* \to A^*
\end{displaymath}
where $e: 1 \to A^*$ is the identity and $m|: A \times A^* \to A^*$ is the restriction of the monoid multiplication along the evident inclusion $i \times 1: A \times A^* \to A^* \times A^*$.

This ``fixed point'' of $F$ can be thought of as the result of the (slightly nonsensical) calculation

\begin{displaymath}
1 + A \times X = X \Rightarrow X = \frac1{1 - A} = 1 + A + A^2 + \ldots = A^*
\end{displaymath}
which can be made rigorous by interpreting the initial equality as defining the solution $X$ by recursion, and applying the theorem above.


\item Let $F: Set \to Set$ be the functor $F(X) = 1 + X^2$. Then the initial $F$-algebra is the set $T$ of isomorphism classes of finite (planar, rooted) binary [[trees]]. The $F$-algebra structure is

\begin{displaymath}
(r, j): 1 + T^2 \to T
\end{displaymath}
where $r: 1 \to T$ names the tree consisting of just a root vertex, and $j: T^2 \to T$ creates a tree $t \vee t'$ from two trees $t$, $t'$ by joining their roots to a new root, so that the root of $t$ becomes the left child and the root of $t'$ the right child of the new root.

The recursive equation

\begin{displaymath}
T = 1 + T^2
\end{displaymath}
would seem to imply that the structure $T$ behaves something like a structural ``sixth root of unity'', and indeed the structural isomorphism $T \cong F(T)$ allows one to exhibit an isomorphism

\begin{displaymath}
T = T^7
\end{displaymath}
constructively, as famously explored in the paper by Andreas Blass, [[seven trees in one]].


\item Let $F: Set \to Set$ be the functor $F(X) = X^*$ (the free monoid from an earlier example). Then the initial $F$-algebra is the set of isomorphism classes of finite planar rooted [[trees]] (not necessarily binary as in the previous example).


\item Let $C$ be the [[coslice category]] $\mathbb{Z} \downarrow Ab$, and let $F: C \to C$ be the functor which pushes out along the multiplication-by-$p$ map $p \cdot -: \mathbb{Z} \to \mathbb{Z}$. Then the initial $F$-algebra is the [[Pruefer group]] $\mathbb{Z}[p^{-1}]/\mathbb{Z}$. See the discussion at the n-Category Caf\'e{}, starting \href{http://golem.ph.utexas.edu/category/2008/11/coalgebraically_thinking.html#c020660}{here}.


\item Let $Ban$ be the category of complex [[Banach spaces]] and maps of [[norm]] bounded above by $1$, and let $F: \mathbb{C} \downarrow Ban \to \mathbb{C} \downarrow Ban$ be the squaring functor $X \mapsto X \times X$. Then the initial $F$-algebra is $L^1([0, 1])$ (with respect to the usual [[Lebesgue measure]]). This result is due to [[Tom Leinster]]; see this \href{http://mathoverflow.net/questions/23143/what-theorem-constructs-an-initial-object-for-this-category-formerly-integrabi/78262#78262}{MathOverflow discussion}.



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[inductive type]], \textbf{initial algebra of an endofunctor}


\item [[higher inductive type]], [[initial algebra of a presentable ∞-monad]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A textbook account of the basic theory is in \href{http://www.andrew.cmu.edu/course/80-413-713/notes/chap10.pdf}{chapter 10} of

\begin{itemize}%
\item [[Steve Awodey]], \emph{Category theory} lecture notes (2011) (\href{http://www.andrew.cmu.edu/course/80-413-713/}{web})

\end{itemize}
A brief review of some basics with an eye towards [[inductive types]] is in section 2 of

\begin{itemize}%
\item [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], \emph{Inductive types in homotopy type theory} (\href{http://arxiv.org/abs/1201.3898}{arXiv:1201.3898})

\end{itemize}
Discussion for [[homotopy type theory]] is in

\begin{itemize}%
\item [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], \emph{Homotopy-initial algebras in type theory} (\href{http://arxiv.org/abs/1504.05531}{arXiv:1504.05531})

\end{itemize}
The relation to [[free monads]] is discussed in

\begin{itemize}%
\item \href{http://maciejcs.wordpress.com/}{Maciej}, \emph{\href{http://maciejcs.wordpress.com/2012/04/17/free-monads-and-their-algebras/}{Free monads and their algebras}}

\end{itemize}
[[!redirects initial algebras of an endofunctor]] [[!redirects initial algebras of endofunctors]]

[[!redirects initial algebra for an endofunctor]] [[!redirects initial algebras for an endofunctor]] [[!redirects initial algebras for endofunctors]]

[[!redirects initial algebra over an endofunctor]] [[!redirects initial algebras over an endofunctor]] [[!redirects initial algebras over endofunctors]]



\end{document}