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\section*{simplicial model for weak omega-categories}

\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{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

While an [[algebraic definition of higher category|algebraic]] definition of [[strict ∞-categories]] is comparatively straightforward, algebraic definitions of fully weak $\omega$-categories (aka $\infty$-[[infinity-category|categories]]) are difficult to work with (although some definitions exist, such as those of [[Batanin ∞-category|Batanin]], [[Trimble n-category|Trimble]], [[Leinster n-category|Leinster]], and [[n-category|Penon]]).

However, just like strict $\omega$-categories have a [[simplicial set|simplicial]] [[nerve]] -- a [[complicial set]] -- induced by the [[orientals]], and just like the category of strict $\omega$-categories is equivalent to the category of complicial sets, one expects that every weak $\omega$-category naturally has a simplicial nerve and that the theory of algebraically defined weak $\omega$-categories is equivalent to the theory of the simplicial sets that arise as their nerves. In fact, one can hope that the theory is \emph{simpler} in the weak case: complicial sets are simplicial sets equipped with extra [[stuff, structure, property|structure]] (a [[stratified simplicial set|stratification]] representing the chosen `strict' composites), while in the nerve of a weak $\omega$-category this structure might be reducible to a property.

In the context of \emph{simplicial models for weak $\omega$-categories} the goal is to characterize those [[simplicial sets]] (or [[stratified simplicial sets]]) which should arise as [[nerves]] of algebraically defined weak $\omega$-categories and thus provide a [[geometric definition of higher category]] generalizing the familiar simplicial description of $\omega$-[[infinity-groupoid|groupoids]] as [[Kan complexes]] and of $(\omega,1)$-[[(infinity,1)-category|categories]] as [[quasi-category|quasi-categories]] to general higher categories.

In effect, the goal is to \emph{define} a weak $\omega$-category to be a certain sort of ([[stratified simplicial set|stratified]]) simplicial set. One could then hope to prove that these are precisely the [[nerve|nerves]] of weak $\omega$-categories defined in some other way.

To distinguish the study of weak $\omega$-categories in terms of their presumed nerves from the study of their would-be algebraic descriptions people speak of [[simplicial weak omega-category|simplicial weak ∞-categories]], see in particular the articles by Dominic Verity referenced below. One should just beware that in this context [[simplicial weak ∞-category]] is not meant as [[simplicial object]] in the category of weak $\omega$-categories.

This program was originally begun by Ross Street and has been carried forward by Dominic Verity with the theory of [[weak complicial sets]]. It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra ``saturation'' condition ensuring that ``every [[equivalence]] is [[thin element|thin]].''

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

\begin{itemize}%
\item [[Dominic Verity]], \emph{Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory} (\href{http://arxiv.org/abs/math.CT/0604414}{arXiv})


\item [[Dominic Verity]], \emph{Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-categories} (\href{http://arxiv.org/PS_cache/math/pdf/0604/0604416v1.pdf}{pdf})



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
Discussion on a previous version of this entry:

[[Mike Shulman|Mike]]: This term is kind of unfortunate; \emph{simplicial weak $\omega$-category} could also mean a simplicial object in weak $\omega$-categories. I don't suppose we can do anything about that?

[[Urs Schreiber|Urs]]: my impression is that what Dominic Verity mainly wants to express with the term is ``simplicial model for weak $\omega$-category''. Maybe we could/should use a longer phrase like that?

[[Mike Shulman|Mike]]: That would make me happier.

[[Urs Schreiber|Urs]]: okay, I changed it. Let me know if this is good now.

\emph{Toby}: But what about `globular $\omega$-category' and things like that? Doesn't `simplicial $\omega$-category' fit right into that framework? This page title sounds like an entire framework for defining $\omega$-category rather than a single $\omega$-category simplicially defined.

[[Urs Schreiber|Urs]]: i am open to suggestions -- but notice that it does indeed seem to me that Dominic Verity wants to express ``an entire framework for defining $\omega$-category'', namely the framework where one skips over the attempt to define $\omega$-categories and instead tries to find a characterization of what should be their nerves.

\emph{Toby}: OK, that fits in with most of what's written here, but not the beginning

\begin{quote}%
\textbf{Simplicial models for weak $\omega$-categories} -- sometimes called [[simplicial weak omega-category|simplicial weak ∞-categories]] -- are \ldots{} Maybe that was just poorly written, but it threw me off. Should it be A \textbf{simplicial model for weak $\omega$-categories} -- which are then sometimes called [[simplicial weak ∞-categories]] -- is \ldots{} or even A \textbf{simplicial model for weak $\omega$-categories} is \ldots{} and only later mention [[simplicial weak ∞-categories]]?


\end{quote}
[[Mike Shulman|Mike]]: You're right that `simplicial $\omega$-category' it fits into `globular $\omega$-category' and `opetopic $\omega$-category' and so on. It seems more problematic in this case, though, since simplicial objects of random categories are a good deal more prevalent than globular ones and opetopic ones. But perhaps I should just live with it.

[[Urs Schreiber|Urs]]: I have now expanded the entry text on this point, trying to make very clear to the reader what's going on here.

\emph{Toby}: Thanks, that's much clearer. And if Verity's definition is at [[weak complicial set]], then we may not really need anything at [[simplicial weak ∞-category]], so no need to offend Mike's sensibilities ({\tt \symbol{94}}\_{\tt \symbol{94}}) either.

[[!redirects simplicial model for weak ∞-categories]] [[!redirects simplicial models for weak ∞-categories]] [[!redirects simplicial models for weak omega-categories]]



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