\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*{free operad}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{explicit_construction}{Explicit construction}\dotfill \pageref*{explicit_construction} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{rooted_tree_operad}{Rooted tree operad}\dotfill \pageref*{rooted_tree_operad} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{free [[operad]]} is [[free functor|free]] on a collection of operations.

Given a collection $k$-ary operations-to-be for each $k \in \mathbb{N}$, the free operad on this collection has as $n$-ary operations the collection of all [[trees]] with $n$ leaves equipped with a labelling of each vertex $v$ with a $k$-ary operation, for $k$ the incoming edges to $v$.

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

Let $V$ be a [[symmetric monoidal category]].

For $G$ a [[discrete group]], write $V^G$ for the category of objects of $V$ equipped with a $G$-[[action]]. For $V$ symmetric monoidal this is again a [[symmetric monoidal category]] and the [[forgetful functor]] $V^G \to V$ is symmetric monoidal.

\begin{defn}
\label{}\hypertarget{}{}
The category of \textbf{collections} (\hyperlink{BergerMoerdijk}{Berger-Moerdijk}) or \textbf{$\mathbb{S}$-modules} (\hyperlink{GetzlerKapranov}{Getzler-Kapranov}) of $V$, or the category of $V$-[[species]], is

\begin{displaymath}
V Coll := \prod_{n \in \mathbb{N}} V^{S_n}
  \,.
\end{displaymath}
\end{defn}
Notice that both $S_0$ and $S_1$ are the trivial group.

So a $V$-operad $P$ is a special $V$-collection with extra structure relating its components. This gives an evident [[forgetful functor]]

\begin{displaymath}
U : V Operad \to V Coll
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
The [[free functor]] [[left adjoint]] to this forgetful functor is the the \textbf{free operad functor}

\begin{displaymath}
F : V Coll \stackrel{\leftarrow}{\to} V Operad : U
  \,.
\end{displaymath}
For $C$ a given collection, we call $F(C)$ the operad \emph{free on the collection $C$}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This free/forgetful [[adjunction]] is used to define the [[model structure on operads]] by [[transferred model structure|transfer]].

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

\hypertarget{explicit_construction}{}\subsubsection*{{Explicit construction}}\label{explicit_construction}

The free operad functor may more explcitly be described as follows (see (\hyperlink{BergerMoerdijk}{Berger-Moerdijk, section 5.8})).

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathbb{T} := Core(\Omega_pl)$ be the [[core]] of the category of planar rooted [[trees]] and \emph{non-planar} morphisms (so the morphisms need not respect the given planar structure).

Write

\begin{itemize}%
\item $t_n \in \Omega_n$ for the $n$-corolla (the tree with a single vertex, $n$ inputs and its unique output root);


\item for $T$ any tree with $n$-ary root vertex let $\{T_i\}_{i=1}^n$ be the sub-trees such that $T = t_n \circ (T_1, \cdots, T_n)$.



\end{itemize}
Then every collection $K \in V Coll$ defines a functor $\bar K : \mathbb{T}^{op} \to V$ by the inductive formula

\begin{displaymath}
\bar K (|) \coloneqq I
\end{displaymath}
\begin{displaymath}
\bar K(t_n(T_1, \cdots, T_n))
  \coloneqq K(n) \otimes \bar K(T_1) \otimes \cdots \bar K(T_n)
  \,.
\end{displaymath}
Define moreover the functor

\begin{displaymath}
\lambda : \mathbb{T} \to Set
\end{displaymath}
to be the functor that sends a tree to the set of numberings of its leaves, and let $\bar \lambda : \mathbb{T} \to V$ be given by postcomposition with $S \mapsto \coprod_{s \in S} I$, where on the right we have the [[coproduct]] of ${\vert S \vert}$ copies of the tensor unit in the [[monoidal category]] $V$.

\end{defn}
So for $T$ a tree with $n$ leaves we have

\begin{displaymath}
\bar \lambda(T) \simeq \coprod_{\sigma \in \Sigma_n} I
  \,,
\end{displaymath}
where the coproduct ranges over the elements of the [[symmetric group]] on $n$ elements.

\begin{prop}
\label{}\hypertarget{}{}
The \textbf{free operad} on a collection $K$ is isomorphic to the [[coend]]

\begin{displaymath}
\bar K \otimes_{\mathbb{T}} \bar \lambda
  =
  \int^{T \in \mathbb{T}}
   \bar K(T) \otimes \bar \lambda(T)
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
The groupoid $\mathbb{T}$ is equivalent to the [[disjoint union]] over isomorphism classes of planar trees of the one-object groupoids with morphisms the given [[automorphism group]]

\begin{displaymath}
\mathbb{T} \simeq \coprod_{[T] \in \pi_0\mathbb{T}}
  \mathbf{B} Aut(T)
  \,.
\end{displaymath}
Therefore the above coend is equivalently

\begin{displaymath}
\bar K \otimes_{\mathbb{T}} \bar \lambda
  =
  \coprod_{[T] \in \pi_0\mathbb{T}}
   \bar K(T) \otimes_{Aut(T)} \bar \lambda(T)
  \,.
\end{displaymath}
\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{rooted_tree_operad}{}\subsubsection*{{Rooted tree operad}}\label{rooted_tree_operad}

Let $K$ be the collection with $K(0) = \emptyset$ and $K(n) = I$ for $n \gt 0$. The corresponding free operad has as $n$-ary operations all rooted trees with $n$ leaves. And composition of operations is given by grafting of trees.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[free category]], [[free monoid]]

\end{itemize}
Riemann surfaces operad (TO BE EXPANDED)

Deligne-Mumford opeard (TO BE EXPANDED)

[[little discs operad]], [[framed little discs operad]] (TO BE EXPANDED) -- See [[Deligne conjecture]]

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

A brief remark on free operads is in (1.12) of

\begin{itemize}%
\item [[Ezra Getzler]], [[Mikhail Kapranov]], \emph{Cyclic operads and cyclic homology}, Conf. Proc. Lect. Notes Geom. Topology IV, Int. Press Camb. (1995), 167--201.

\end{itemize}
A detailed discussion is in Part II, chapter I, section 1.9 of

\begin{itemize}%
\item [[Martin Markl]], S. Shnider, [[Jim Stasheff]], \emph{Operads in Algebra, Topology and Physics}, Surveys and Monographs of the Amer. Math. Soc. 96 (2002).

\end{itemize}
and in section 3 of

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{The Boardman-Vogt resolution of operads in monoidal model categories} , Topology 45 (2006), 807--849. (\href{http://math.unice.fr/~cberger/BV.pdf}{pdf})

\end{itemize}
The [[coend]]-description is given in section 5.8 of

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Axiomatic homotopy theory for operads} (\href{http://arxiv.org/abs/math/0206094v3}{arXiv:0206094})

\end{itemize}
[[!redirects collection]] [[!redirects collections]]

[[!redirects free operads]]



\end{document}