\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*{multicategory}

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

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - 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{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{in_terms_of_cartesian_monads}{In terms of cartesian monads}\dotfill \pageref*{in_terms_of_cartesian_monads} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_operads}{Relation to operads}\dotfill \pageref*{relation_to_operads} \linebreak
\noindent\hyperlink{relation_to_monoidal_categories}{Relation to monoidal categories}\dotfill \pageref*{relation_to_monoidal_categories} \linebreak
\noindent\hyperlink{examples_and_special_cases}{Examples and special cases}\dotfill \pageref*{examples_and_special_cases} \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}

Recall that a [[category]] consists of a collection of [[morphisms]] each having a single [[object]] as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A \emph{multicategory} is like a category, except that one allows multiple inputs and a single output.

Another term for \emph{multicategory} is \emph{coloured [[operad]]}.

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

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

A \textbf{multicategory} $C$ consists of

\begin{itemize}%
\item A collection of \emph{objects}, $C_0$.
\item A collection of \emph{multimorphisms}, $C_1$.
\item A source map $s: C_1 \to (C_0)*$ to the collection of finite, possibly empty [[list]]s of objects (where $(C_0)*$ is the [[free monoid]] generated by $C_0$), and a target map $t: C_1 \to C_0$. We write $f: c_1, \ldots, c_n \to c$ to indicate the source and target of a multimorphism $f$.
\item Identity and composition laws. The identity law is a map $1_{-}: C_0 \to C_1$ where $1_c: c \to c$. The composition law assigns, to each $f: c_1, \ldots, c_n \to c$ together with an $n$-tuple $\langle f_i: \vec{c}_i \to c_i: i = 1, \ldots, n \rangle$, a composite\begin{displaymath}
f \circ (f_1, \ldots, f_n): \vec{c}_1, \ldots, \vec{c}_n \to c
\end{displaymath}
where the source is obtained by concatenating lists in the evident way.



\end{itemize}
These operations are subject to [[associativity]] and [[identity]] axioms which the reader can probably figure out, but see for example (\hyperlink{Leinster}{Leinster, page 35 ff.}), for details.

Many people (especially non-category theorists) use the word \emph{multicategory} or the word \emph{colored [[operad]]} to mean what we would call a \emph{[[symmetric multicategory]]} / \emph{[[symmetric operad]]}. These are multicategories equipped with an [[action]] of the [[symmetric group]] $S_n$ on the multimorphisms $c_1, \ldots, c_n \to c$ such that the composition is equivariant with respect to these actions.

\hypertarget{in_terms_of_cartesian_monads}{}\subsubsection*{{In terms of cartesian monads}}\label{in_terms_of_cartesian_monads}

An efficient abstract method for defining multicategories and related structures is through the formalism of [[cartesian monads]]. For ordinary categories, one uses the identity monad on [[Set]]; for ordinary multicategories, one uses the [[free monoid]] monad $(-)*: Set \to Set$. This is a special case of the yet more general notion of [[generalized multicategory]].

We summarize here how the theory applies to the case of a [[cartesian monad]] $T$ on a category with [[pullbacks]]; see [[generalized multicategory]] for the fully general context.

\begin{itemize}%
\item First, a \textbf{$T$-span} from $X$ to $Y$ is a [[span]] $p$ from $T X$ to $Y$, that is, a diagram\begin{displaymath}
T X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y
\end{displaymath}
A $T$-span is often written as $p: X &#8696; Y$.



\end{itemize}
When $T$ is the free monoid monad on $Set$, a $T$-span from $X$ to itself is called a \emph{multigraph} on $X$.

\begin{itemize}%
\item $T$-spans are the 1-cells of a [[bicategory]]. A 2-cell between $T$-spans $e, f: X &#8696; Y$ is a 2-cell between ordinary spans from $T X$ to $Y$. To horizontally compose $T$-spans $e: X &#8696; Y$ and $f: Y &#8696; Z$, take the ordinary span composite of

\begin{displaymath}
(T X \stackrel{m X}{\leftarrow} T^2 X \stackrel{T e_1}{\leftarrow} T E \stackrel{T e_2}{\to} T Y) \circ (T Y \stackrel{f_1}{\leftarrow} F \stackrel{f_2}{\to} Z)
\end{displaymath}
where $m: T^2 \to T$ is the monad multiplication. The identity $T$-span from $X$ to itself is the span

\begin{displaymath}
T X \stackrel{u X}{\leftarrow} X \stackrel{1_X}{\to} X
\end{displaymath}
where $u: I \to T$ is the monad unit. The verification of the bicategory axioms uses the cartesianness of $T$ in concert with the corresponding axioms on the bicategory of spans.


\item A \emph{$T$-multicategory} is defined to be a [[monad]] in the bicategory of $T$-spans.



\end{itemize}
When $T$ is the free monoid monad on sets, then a $T$-multicategory is a multicategory as defined above. For more examples and generalizations, see [[generalized multicategory]].

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

\hypertarget{relation_to_operads}{}\subsubsection*{{Relation to operads}}\label{relation_to_operads}

A [[nonpermutative operad|nonpermutative]] (or Stasheff-) [[operad]] in [[Set]] may be defined as an ordinary multicategory with exactly one object. Likewise, a [[symmetric operad]] in any [[symmetric monoidal category]] $V$ is equivalent to a $V$-[[enriched category|enriched]] multicategory with one object.

More generally, the notion of \emph{multi-colored [[planar operad]]} is equivalent to that of multicategory, and the notion of \emph{multi-colored [[symmetric operad]]} is equivalent to that of [[symmetric multicategory]].

Fully generally, for each cartesian monad $T$, there is a corresponding notion of $T$-operad, namely a $T$-multicategory whose underlying $T$-span has the form $1 &#8696; 1$.

For example, in Batanin's approach to (weak) $\infty$-[[infinity-category|categories]], a [[globular operad]] is a $T$-operad, where $T$ is the free (strict) $\omega$-[[strict omega-category|category]] monad on the category of [[globular set]]s.

Ordinary (permutative/symmetric) operads, and their generalization to [[symmetric multicategory|symmetric multicategories]], can also be treated in the framework of [[generalized multicategories]], but they require a framework more general than that of cartesian monads.

\hypertarget{relation_to_monoidal_categories}{}\subsubsection*{{Relation to monoidal categories}}\label{relation_to_monoidal_categories}

There is a [[faithful functor]] from [[monoidal categories]] to [[multicategories]], given by forming [[representable multicategories|represented multicategories]]. Conversely, to any multicategory $C$ there is an associated (strict) monoidal category $F(C)$, whose objects (respectively, arrows) are [[lists]] of objects (respectively, arrows) of $C$, and where the tensor product in $F(C)$ is given by concatenation.

\hypertarget{examples_and_special_cases}{}\subsection*{{Examples and special cases}}\label{examples_and_special_cases}

\begin{itemize}%
\item [[permutative categories]] with multi-linear functors between them form a multicategory [[PermCat]]

\end{itemize}
See also the examples at \emph{[[operad]]}.

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

\begin{itemize}%
\item [[multimorphism]]


\item [[multifunctor]]


\item [[polycategory]]


\item [[fibration of multicategories]]


\item [[symmetric multicategory]]


\item [[generalized multicategory]], [[operad]]



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

\begin{itemize}%
\item [[Tom Leinster]], \emph{Higher operads, higher categories}, London Math. Soc. Lec. Note Series \textbf{298}, \href{http://arxiv.org/abs/math.CT/0305049}{math.CT/0305049}

\end{itemize}
[[!redirects multicategories]]

[[!redirects pseudomonoidal category]]



\end{document}