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\begin{document}

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\section*{weak complicial set}

\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{model_structure}{Model structure}\dotfill \pageref*{model_structure} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Weak complicial sets are [[simplicial sets]] with [[stuff, structure, property|extra structure]] that are closely related to the [[∞-nerve]]s of weak [[∞-categories]].

The goal of characterizing such nerves, without an \emph{a priori} definition of ``weak $\omega$-category'' to start from, is called [[simplicial weak ∞-category]] theory. 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]].'' General weak complicial sets can be regarded as ``presentations'' of weak $\omega$-categories.

Weak complicial sets are a joint generalization of

\begin{itemize}%
\item [[strict omega-category|strict ∞-categories]];


\item [[Kan complexes]];


\item [[quasi-category|quasi-categories]].



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

\begin{defn}
\label{}\hypertarget{}{}
Let

\begin{itemize}%
\item $\Delta^k[n]$ be the [[stratified simplicial set]] whose underlying simplicial set is the $n$-[[simplex]] $\Delta[n]$, and whose marked cells are precisely those simplices $[r] \to [n]$ that contain $\{k-1, k, k+1\} \cap [n]$;


\item $\Lambda^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $k$-[[horn]] of $\Delta[n]$, with marked cells those that are marked in $\Delta^k[n]$;


\item $\Lambda^k[n]'$ be obtained from $\Delta^k[n]$ by making the $(k-1)$st $(n-1)$-face and the $(k+1)$st $(n-1)$ face thin;


\item $\Delta^k[n]''$ be obtained from $\Delta^k[n]$ by making all $(n-1)$-faces thin.



\end{itemize}
An \textbf{elementary anodyne extension} in $Strat$, the category [[stratified simplicial sets]] is

\begin{itemize}%
\item a \textbf{complicial horn extension} $\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$

\end{itemize}
or

\begin{itemize}%
\item a \textbf{complicial thinness extension} $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$

\end{itemize}
for $n = 1,2, \cdots$ and $k \in [n]$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A [[stratified simplicial set]] is a \textbf{weak complicial set} if it has the [[right lifting property]] with respect to all

$\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$ and $\Lambda^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$

A [[complicial set]] is a weak complicial set in which such liftings are unique.

\end{defn}
\hypertarget{model_structure}{}\subsection*{{Model structure}}\label{model_structure}

There is a [[model category]] structure that presents the [[(infinity,1)-category]] of weak complicial sets, hence that of weak $\omega$-categories. See

\begin{itemize}%
\item [[model structure for weak complicial sets]].

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item For $C$ a [[strict ∞-category]] and $N(C)$ its [[oriental|∞-nerve]], the \emph{Roberts stratification} which regards each identity morphism as a thin cell makes $N(C)$ a strict [[complicial set]], hence a weak complicial set. This example is not ``saturated.''


\item There is also the [[stratified simplicial set|stratification]] of $N(C)$ which regards each $\omega$-equivalence morphism as a thin cell. $N(C)$ with this stratification is a weak complicial set (example 17 of \href{http://arxiv.org/abs/math/0604414}{Ver06}). This should be the ``saturation'' of the previous example, and exhibits the inclusion of strict $\omega$-categories into weak ones.


\item A simplicial set is a weak complicial set when equipped with its maximal [[stratified simplicial set|stratification]] (every simplex of dimension $\gt 0$ is thin) if and only if it is a [[Kan complex]]. This example is, of course, saturated, and is viewed as embedding $\omega$-groupoids into $\omega$-categories.


\item A simplicial set is a [[quasi-category]] if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension $\gt 1$ is thin, and only degenerate 1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its [[homotopy category]]). It presents the embedding of $(\infty,1)$-[[(infinity,1)-category|categories]] into weak $\omega$-categories.

Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why $QCat$ can ``correctly'' be regarded as a full subcategory of $sSet$. This is not true at higher levels; for instance not every simplicial map between nerves of strict $\omega$-categories necessarily preserves $\omega$-equivalence morphisms.



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

The definition of weak complicial sets is definition 14, page 9 of

\begin{itemize}%
\item [[Dominic Verity]], \emph{Weak complicial sets Part I: Basic homotopy theory} (\href{http://arxiv.org/abs/math/0604414}{arXiv})

\end{itemize}
Further developments are in

\begin{itemize}%
\item Dominic Verity, \emph{Weak complicial sets Part II: Nerves of complicial Gray-categories} (\href{http://arxiv.org/abs/math/0604416}{arXiv})

\end{itemize}
A model structure for [[(infinity,n)-categories]] is presented in

\begin{itemize}%
\item [[Viktoriya Ozornova]], Martina Rovelli, Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces, \href{https://arxiv.org/abs/1809.10621}{arxiv}

\end{itemize}
[[!redirects weak complicial sets]] [[!redirects weak compicial set]]



\end{document}