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\begin{document}

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\section*{strict 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{as_simplicial_sets}{As simplicial sets}\dotfill \pageref*{as_simplicial_sets} \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{strict $\omega$-category} is a [[globular set|globular]] [[∞-category]] in which all operations obey their respective laws strictly.

This was the original notion of [[∞-category]], and the original meaning of the term [[∞-category]]. Even today, most authors who use that term still mean this notion.

This means that

\begin{enumerate}%
\item all composition operations are strictly associative;


\item all composition operations strictly commute with all others (strict [[exchange laws]]);


\item all identity $k$-morphisms are strict identities under all compositions.



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

An \textbf{$\omega$-category} $C$ [[internalization|internal to]] $Sets$ is

\begin{itemize}%
\item a [[globular set]]

\begin{displaymath}
C :=
(\cdots
C_3 \stackrel{\to}{\to}
C_2 \stackrel{\to}{\to}
C_1 \stackrel{\to}{\to}
C_0
)
\end{displaymath}

\item together with the structure of a [[category]] on all $(  C_{k} \stackrel{\to}{\to} C_l )$ for all $k \gt l$;


\item such that $(  C_{k} \stackrel{\to}{\to}
C_{l} \stackrel{\to}{\to} C_m )$ for all $k \gt l \gt m$; is a [[strict 2-category]].



\end{itemize}
Similarly for an $\omega$-category [[internalization|internal to]] another ambient category $A$.

The category $\omega Cat(A)$ of $\omega$-categories [[internalization|internal to]] $A$ has $\omega$-categories as its objects and morphism of the underlying globular objects respecting all the above extra structure as morphisms.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item The last condition in the above definition says that all pairs of composition operations satisfy the [[exchange law]].


\item $\omega$-Categories can also be understood as the directed limit of the sequence of iterated [[enriched category theory|enrichments]]

\begin{displaymath}
(0 Cat = Set)
\hookrightarrow
(1 Cat = Set-Cat)
\hookrightarrow
(2 Cat = Cat-Cat)
\hookrightarrow
\left(3 Cat = (2Cat)-Cat = (Cat-Cat)-Cat\right)
\hookrightarrow
\cdots
\,.
\end{displaymath}

\item The category of strict $\omega$-categories admits a [[closed monoidal category|biclosed monoidal structure]] called the [[Crans-Gray tensor product]].


\item The category of strict $\omega$-categories also admits a [[canonical model structure]].


\item Terminology on $\omega$-categories varies. We here follow section 2.2 of [[Sjoerd Crans]]: \href{http://home.tiscali.nl/secrans/papers/thpp.html}{Pasting presentations for $\omega$-categories}, where it says

\begin{itemize}%
\item \emph{[[Ross Street|Street]] allowed $\omega$-categories to have infinite dimensional cells. Steiner has the extra condition that every cell has to be finite dimensional, and called them $\infty$-categories, following [[Ronnie Brown|Brown]] and Higgins. I will use Steiner's approach here since that's the one that reflects the notion of higher dimensional homotopies closest, but I will nonethless call them $\omega$-categories, and I agree with [[Dominic Verity|Verity]]`s suggestion to call the other ones $\omega^+$-categories.}

\end{itemize}

\item [[Simpson's conjecture]], a statement about [[semi-strict infinity-category|semi-strictness]], states that every weak $\infty$-category should be equivalent to an $\infty$-category in which strictness conditions 1. and 2. hold, but not 3.



\end{itemize}
\hypertarget{as_simplicial_sets}{}\subsection*{{As simplicial sets}}\label{as_simplicial_sets}

Under the [[oriental|∞-nerve]]

\begin{displaymath}
N : Str \omega Cat \to SSet
\end{displaymath}
strict $\omega$-categories yield simplicial sets that are called [[complicial sets]].

\begin{uprop}
The categories of $\omega$-categories and complicial sets are [[equivalence of categories|equivalent]].

\end{uprop}
This is sometimes called the \emph{Street-Roberts conjecture}. It was completely proven in

\begin{itemize}%
\item Dominic Verity, \emph{Complicial sets} (\href{http://arxiv.org/abs/math/0410412}{arXiv})

\end{itemize}
which also presents the history of the conjecture.

Based on this fact, there are attempts to weaken the condition on a [[simplicial set]] to be a [[complicial set]] so as to obtain a notion of [[simplicial weak ∞-category]].

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

\begin{itemize}%
\item [[strict ∞-groupoid]]


\item [[parity complex]], [[oriental]]


\item [[complicial set]]



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

Strict $\omega$-categories have probably been independently invented by several people.

According to \hyperlink{Street09}{Street 09, p. 10} the concept was first brought up by [[John Roberts]] 1977-1978, in an attempt to define [[non-abelian cohomology]] (of [[local nets of observables]] in [[algebraic quantum field theory]]).

Possibly the earliest published definition is due to

\begin{itemize}%
\item [[Ronnie Brown]] and P.J. Higgins, \emph{The equivalence of $\infty$-groupoids and crossed complexes}, Cah. Top. G\'e{}om. Diff. 22 (1981) no. 4, 371-386 \href{http://www.numdam.org/item?id=CTGDC_1981__22_4_371_0}{web}.

\end{itemize}
which also contains the definitions of [[n-fold category]] and of what was later called [[globular set]]. There these strict, globular higher categories are called ``$\infty$-categories'' while ``$\omega$-groupoid'' is used to mean a cubical set with connections and compositions, each a groupoid, as in

\begin{itemize}%
\item R. Brown and P.J. Higgins, \emph{On the algebra of cubes}, J. Pure Appl. Algebra 21 (1981) 233-260.

\end{itemize}
Applications to homotopy theory were given in

\begin{itemize}%
\item R. Brown and P.J. Higgins, \emph{Colimit theorems for relative homotopy groups}, J. Pure Appl. Algebra 22 (1981) 11-41.


\item R. Brown, \emph{Non-abelian cohomology and the homotopy classification of maps}, in Homotopie alg\'e{}brique et algebre locale, Conf. Marseille-Luminy 1982, ed. J.-M. Lemaire et J.-C. Thomas, Ast\'e{}risques 113-114 (1984), 167-172.



\end{itemize}
Related monoidal closed structures were developed in:

\begin{itemize}%
\item R. Brown and P.J. Higgins, \emph{Tensor products and homotopies for $\omega$-groupoids and crossed complexes}, J. Pure Appl. Alg. 47 (1987) 1-33.

\end{itemize}
Another 1980s reference is

\begin{itemize}%
\item [[Ross Street]], \emph{The algebra of oriented simplices}, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 (\href{http://www.math.mq.edu.au/~street/aos.pdf}{pdf}),

\end{itemize}
in which strict $\omega$-categories are called ``$\omega$-categories.'' This paper was also the first to define [[orientals]].

A review of some of the theory in the context of some of the history is given in

\begin{itemize}%
\item [[Ross Street]], \emph{An Australian conspectus of higher categories}, chapter in \emph{Towards Higher Categories} Volume 152 of the series The IMA Volumes in Mathematics and its Applications pp 237-264 (\href{http://www.math.uchicago.edu/~may/IMA/Street.pdf}{pdf})

\end{itemize}
and also in

\begin{itemize}%
\item [[Ross Street]], \emph{Categorical and combinatorial aspects of descent theory} (\href{http://arxiv.org/abs/math.CT/0303175}{arXiv})

\end{itemize}
The theory of $\omega$-categories was further developed by Sjoerd Crans in parts 2 and 3 of his \href{http://home.tiscali.nl/secrans/papers/comb.html}{thesis}

\begin{itemize}%
\item [[Sjoerd Crans]], \emph{Pasting presentations for $\omega$-categories} (\href{http://home.tiscali.nl/secrans/papers/thpp.html}{link})


\item Sjoerd Crans, \emph{Pasting schemes for the monoidal biclosed structure on $\omega$-Cat} (\href{http://home.tiscali.nl/secrans/papers/thten.html}{link})



\end{itemize}
See also the

\begin{itemize}%
\item \href{http://home.tiscali.nl/secrans/papers/thintro.ps.gz}{Introduction}

\end{itemize}
to his thesis, in particular section I.3 ``$\omega$-categories''.

The relationship between strict $\omega$-categories and cubical $\omega$-categories was considered in

\begin{itemize}%
\item F.A. Al-Agl, R. Brown, R. Steiner \emph{Multiple categories: the equivalence of a globular and a cubical approach}, Adv. Math. 170 (2002), no. 1, 71--118

\end{itemize}
where they prove that strict globular $\omega$-categories are equivalent to $\omega$-fold categories (aka ``cubical $\omega$-categories'') equipped with [[connections]]. This paper also develops the monoidal closed structures.

\begin{itemize}%
\item R. Steiner, \emph{Omega-categories and chain complexes}, Homology, Homotopy and Applications \textbf{6}(1), 2004, pp.175--200, \href{http://www.intlpress.com/HHA/v6/n1/a12/v6n1a12.pdf}{pdf} [[!redirects strict ∞-category]] [[!redirects strict ∞-categories]] [[!redirects strict omega-categories]]

\end{itemize}


\end{document}