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\begin{document}

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\section*{opetopic omega-category}

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{by_baezdolan}{By Baez-Dolan}\dotfill \pageref*{by_baezdolan} \linebreak
\noindent\hyperlink{by_makkai}{By Makkai}\dotfill \pageref*{by_makkai} \linebreak
\noindent\hyperlink{DefinitionByPalm}{By Palm}\dotfill \pageref*{DefinitionByPalm} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[higher category theory]], a concept of \emph{opetopic omega-category} is one of \emph{[[weak omega-category]]} modeled on [[opetope|opetopic]] [[geometric shape for higher structures|shapes]].

There are various flavors of the definition. Typically they all say that an opetopic omega-category is an [[opetopic set]] equipped with certain [[structure]] and [[property]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{by_baezdolan}{}\subsubsection*{{By Baez-Dolan}}\label{by_baezdolan}

(\hyperlink{BaezDolan97}{Baez-Dolan 97})

\hypertarget{by_makkai}{}\subsubsection*{{By Makkai}}\label{by_makkai}

(\hyperlink{HermidaMakkaiPower01}{Hermida-Makkai-Power 01}, \hyperlink{Makkai}{Makkai})

The actual definition of opetopic $\omega$-categories, called ``multitopic $\omega$-categories'' here, appears on p.57 of (\hyperlink{Makkai}{Makkai}).

\hypertarget{DefinitionByPalm}{}\subsubsection*{{By Palm}}\label{DefinitionByPalm}

In the definition by (\hyperlink{Palm}{Palm}) of opetopic $\omega$-category, the extra [[structure]] on an [[opetopic set]] is that some cells are labeled as ``thin'' (as [[equivalences]]) and the extra [[property]] are two classes of higher dimensional [[horn]]-filler conditions:

\begin{enumerate}%
\item given a [[horn]] (``niche'', ``nook'') obtained from an [[opetope]] by discarding the [[codimension]]-0 interior and the outgoing [[codimension]]-1 face, then there is a filler opetope whose codimension-0 interior is marked as an equivalence. The filling outgoing codimension-1 face is marked an equivalence if all the other codimension-1 faces are.


\item given a [[horn]] (``niche'', ``nook'') obtained from an [[opetope]] by discarding the [[codimension]]-0 interior and an incoming codimension-1 face such that all the remaining codimension-1 faces are marked as equivalences, then there exists a filler both whose codimension-1 and codimension-0 cell are marked as equivalences.



\end{enumerate}
This definition has been given a [[syntax|syntactic]] formulation by (\hyperlink{Finster12}{Finster 12}) in terms of \emph{[[opetopic type theory]]}.

However, there is no saturation condition in this definition.

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

An overview is in \href{http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf#page=63}{chapter 4} of

\begin{itemize}%
\item [[Eugenia Cheng]], [[Aaron Lauda]], \emph{Higher dimensional categories: an illustrated guidebook} (\href{http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf}{pdf})

\end{itemize}
and in chapter 7 of

\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}
Opetopes were introduced here:

\begin{itemize}%
\item [[John Baez]], [[James Dolan]], Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, \emph{Adv. Math.} \textbf{135} (1998), 145--206. (\href{http://arxiv.org/abs/q-alg/9702014}{arXiv:q-alg/9702014})

\end{itemize}
Some mistakes were corrected in subsequent papers:

\begin{itemize}%
\item [[Eugenia Cheng]], The category of opetopes and the category of opetopic sets, \emph{Th. Appl. Cat.} \textbf{11} (2003), 353--374. \href{http://arxiv.org/abs/math/0304284}{arXiv:0304284})


\item [[Tom Leinster]], \emph{Structures in higher-dimensional category theory}, (\href{http://arxiv.org/abs/math/0109021}{arXiv:0109021})



\end{itemize}
Makkai and collaborators introduced a slight variation they called `multitopes':

\begin{itemize}%
\item [[Claudio Hermida]], [[Michael Makkai]], [[John Power]], \emph{On weak higher-dimensional categories I, II} \emph{Jour. Pure Appl. Alg.} \textbf{157} (2001), 221--277 (\href{http://www.sciencedirect.com/science/article/pii/S0022404999001796}{journal}, \href{http://www.math.mcgill.ca/makkai/multitopicsets/}{ps.gz files}, [[HermidaMakkaiPower01.pdf:file]])


\item [[Michael Makkai]], The multitopic $\omega$-category of all multitopic $\omega$-categories. (\href{http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf}{pdf}, \href{http://www.math.mcgill.ca/makkai/mltomcat04/}{web})



\end{itemize}
Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic $n$-categories:

\begin{itemize}%
\item [[Eugenia Cheng]], \emph{Weak $n$-categories: comparing opetopic foundations}, Jour. Pure Appl. Alg. \textbf{186} (2004), 219--231. (\href{http://arxiv.org/abs/math/0304279}{arXiv:0304279})


\item [[Eugenia Cheng]], \emph{Weak $n$-categories: opetopic and multitopic foundations}, Jour. Pure Appl. Alg.\_\textbf{186} (2004), 109--137.(\href{http://arxiv.org/abs/math/0304277}{arXiv:0304277})



\end{itemize}
She has also shown that opetopic [[bicategories]] are ``the same'' as the ordinary kind:

\begin{itemize}%
\item [[Eugenia Cheng]], Opetopic bicategories: comparison with the classical theory. (\href{http://arxiv.org/abs/math/0304285}{arXiv})

\end{itemize}
A higher dimensional [[string diagram]]-notation for opetopes was introduced (as ``zoom complexes'' in section 1.1) in

\begin{itemize}%
\item [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], \emph{Polynomial functors and opetopes} (\href{http://arxiv.org/abs/0706.1033}{arXiv:0706.1033})

\end{itemize}
Animated exposition of this higher-dimensional string-diagram notation is in

\begin{itemize}%
\item [[Eric Finster]], \emph{Opetopic Diagrams 1 - Basics} (\href{http://www.youtube.com/watch?v=OANwLohwJqk}{video})


\item [[Eric Finster]], \emph{Opetopic Diagrams 2 - Geometry} (\href{http://www.youtube.com/watch?v=E7OvuA1jRKM}{video})



\end{itemize}
The variant of \href{opetopic+omega-category#DefinitionByPalm}{Palm opetopic omega-categories} is due to

\begin{itemize}%
\item [[Thorsten Palm]], \ldots{}

\end{itemize}
A [[syntax|syntactic]] formalization of [[opetopic omega-categories]] in terms of [[opetopic type theory]] is in

\begin{itemize}%
\item [[Eric Finster]], \emph{Type Theory and the Opetopes}, talk at \href{http://www.mimuw.edu.pl/~zawado/SemTK/OSTKA.html}{Polish Seminar on Category Theory and its Applications, June 2012} (\href{http://sma.epfl.ch/~finster/opetope/types-and-opetopes.pdf}{pdf})

\end{itemize}
[[!redirects opetopic omega-categories]]

[[!redirects opetopic ∞-category]] [[!redirects opetopic ∞-categories]]



\end{document}