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\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*{Clemens Berger, A Cellular Nerve for Higher Categories} This entry draws from \begin{itemize}% \item [[nLab:Clemens Berger]], \emph{A cellular nerve for higher categories}, Advances in Mathematics 169, 118-175 (2002) (\href{http://math1.unice.fr/~cberger/nerve.pdf}{pdf}) \end{itemize} \hypertarget{0_notation_and_terminology}{}\subsection*{{0. Notation and terminology}}\label{0_notation_and_terminology} \hypertarget{01}{}\subsubsection*{{0.1}}\label{01} \hypertarget{02_higher_graphs_and_higher_categories}{}\subsubsection*{{0.2 Higher graphs and higher categories}}\label{02_higher_graphs_and_higher_categories} \begin{defn} \label{globe}\hypertarget{globe}{} The \emph{globe category} $G$ is defined to be the category with one object in each degree and the \emph{globular operators} $s,t$ are defined by the identities \begin{displaymath} s\circ s=s\circ t \end{displaymath} \begin{displaymath} t\circ t= t\circ s \end{displaymath} A presheaf on $G$ is called a \emph{globular set} or \emph{omega graph} or \emph{$\omega$-graph}. $\omega$-graphs with natural transformations as morphisms form a category denoted by $\omegaGraph$. \end{defn} \begin{defn} \label{}\hypertarget{}{} Let $C$ be $2$-category with underlying reflexive $2$-graph $C_i$ for $i=0,1,2$ with globular operators given by source, target, and identity. Then $(C_i)_i$ comes with three composition laws \begin{displaymath} \circ_i^j: C_j\times_i C_j\to C_j \end{displaymath} for $0\le i\lt j\le 2$. Spelled out this means: $i=0, j=1$: composition of $1$-morphism along $0$-morphisms (i.e.objects) $i=0,j=2$: composition of $2$-morphisms along $0$-morphisms (i.e.objects), also called \emph{[[nLab:horizontal composition]]}. $i=1,j=2$: composition of $2$-morphisms along $1$-morphisms, also called \emph{[[nLab:vertical composition]]}. Then \emph{Godement´s interchange rule} or \emph{Godement´s interchange law} or just \emph{[[nLab:interchange law]]} is the assertion that the immediate diagrams commute. Note that there is on more type of composition of a $1$-morphism with a $2$-morphism called \emph{[[nLab:whiskering]]}. \end{defn} \begin{defn} \label{omega-category}\hypertarget{omega-category}{} An $\omega$-category is defined to be a reflexive graph $(C_i)_i$ such that for every triple $i\lt j\lt k$, the family $(C_i,C_j,C_k;\circ_i^j,\circ_i^k, \circ_j^)$ has the structure of a $2$-category. \end{defn} \hypertarget{1_globular_theories_and_cellular_nerves}{}\subsection*{{1. Globular theories and cellular nerves}}\label{1_globular_theories_and_cellular_nerves} Contents: Batanin's $\omega$-operads are described by their operator categories which are called \emph{globular theories}. \begin{definition} \label{}\hypertarget{}{} A \emph{finite planar level tree} ( or for short just a \emph{tree}) is a graded set $(T(n))_{n\in \mathbb{N}_0}$ endowed with a map $i_T: T_{\gt 0}$ decreasing the degree by one and such that all fibers $i_T^{-1}(x)$ are linearly ordered. The collection of trees with maps of graded sets commuting with $i$ defines a category $\mathcal{T}$, called the \emph{[[nLab:tree category|category of trees]]}. \end{definition} \begin{lemma} \label{}\hypertarget{}{} The finite ordinal $[n]\in \Delta$ we can regard as the 1-level tree with $n$ input edges. Hence the simplex category embeds in the tree category $\Delta\hookrightarrow\mathcal{T}$. \end{lemma} The following ${}_*$-construction is due to Batanin. \begin{lemma} \label{}\hypertarget{}{} Let $T$ be a tree. A \emph{$T$-sector of height $k$} is defined to be a cospan \begin{displaymath} \itexarray{ y^\prime&&y^{\prime\prime} \\ \searrow&&\swarrow \\ &y } \end{displaymath} denoted by $(y;y^\prime,y^{\prime\prime})$ where $y\in T(k)$ and $y\lt y^{\prime\prime}$ are consecutive vertices in the linear order $i_T^{-1}(y)$. The set $GT$ of $T$-sector is graded by the height of sectors. The \emph{source of a sector $(y;y^\prime,y^{\prime\prime})$} is defined to be $(i(y);x,y)$ where $x,y$ are consecutive vertices. The \emph{target of a sector $(y;y^\prime,y^{\prime\prime})$} is defined to be $(i(y);y,z)$ where $y,z$ are consecutive vertices. \begin{displaymath} \itexarray{ &y^\prime&&y^{\prime\prime} \\ & \searrow&&\swarrow \\ x&&y&&z \\ \searrow&&\downarrow^i&&\swarrow \\ &&i(y) } \end{displaymath} To have a source and a target for every sector of $T$ we adjoin in every but the highest degree a lest- and a greatest vertex serving as new minimum and maximum for the linear orders $i^{-1}(x)$. We denote this new tree by $\overline T$ and the set of its sectors by $T_*:=G\overline T(k)$ and obtain \emph{source- and target operators} $s,t:T_*\to T_*$. This operators satisfy \begin{displaymath} s\circ s=s\circ t \end{displaymath} \begin{displaymath} t\circ t =t\circ s \end{displaymath} as one sees in the following diagram depicting an ``augmented'' tree of height $3$ \begin{displaymath} \itexarray{ T(3)&&&y^\prime&&y^{\prime\prime} \\ &&& \searrow&&\swarrow \\ T(2)&&x&&y&&z \\ &&\searrow&&\downarrow^i&&\swarrow \\ T(1)&&u&&v&&w \\ &&\searrow&&\downarrow^i&&\swarrow \\ T(0)&&&&r } \end{displaymath} which means that $T_*$ is an $\omega$-graph (also called [[nLab:globular set]]). Now let $G$ denote the [[nLab:globe category]] whose unique object in degree $n\in \mathbb{N}$ is $n_G$, and let $\mathbf{n}$ denotes the linear $n$-level tree. Then we have $\mathbf{n}_*\simeq G(-,n_G)$ is the standard $n$-globe. (Note that the previous diagram corresponds to the standard $3$ globe.) \end{lemma} \begin{defn} \label{}\hypertarget{}{} Let $f:S_*\to T_*$ be a monomorphism. $f$ is called to be \emph{cartesian} if \begin{displaymath} \itexarray{ (S_*)_n&\stackrel{f_n}{\to}&(T_*)_n \\ \downarrow^s&&\downarrow^t \\ (S_*)_{n-1}&\stackrel{f_{n-1}}{\to}&(T_*)_{n-1} } \end{displaymath} is a pullback for all $n$. \end{defn} \begin{lemma} \label{}\hypertarget{}{} Let $S,T$ be level trees. (1) Any map $S_*\to T_*$ is injective. (2) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to cartesian subobjects of $T_*$. (3) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to plain subtrees of $T$ with a specific choice of $T$-sector for each input vertex of $S$. (\ldots{}) \end{lemma} \begin{definition} \label{}\hypertarget{}{} (1) The category $\Theta_0$ defined by having as objects the level trees and as morphisms the maps between the associated $\omega$-graphs. These morphisms are called \emph{immersions}. This category shall be equipped with the structure of a [[nLab:site]] by defining the covering sieves by epimorphic families (of immersions). This site is called \emph{the globular site}. (2) A \emph{globular theory} is defined to be a category $\Theta_A$ such that \begin{displaymath} \Theta_0\hookrightarrow \Theta_A \end{displaymath} is an inclusion of a wide subcategory such that representable presheaves on $\Theta_A$ restrict to sheaves on $\Theta_0$. (3) Presheaves on $\Theta_A$ which restrict to sheaves on $\Theta_0$ are called $\Theta_A$-models. \end{definition} \begin{lemma} \label{}\hypertarget{}{} The forgetful functor \begin{displaymath} Sh (\Theta_0)\to \omega Graph:=Psh (G) \end{displaymath} is an equivalence of categories. \end{lemma} \begin{proof} Let $X\in Psh(\Theta_0)$ and show that $X\in Sh(\Theta_0)$ iff $X(T)\simeq hom_{Psh(G)}(T_*,X)$ by writing $X$ as a colimit of representables. \end{proof} \begin{defn} \label{}\hypertarget{}{} There is a monad $(w,\eta,\mu)$ on $\omega Graph$ defined by \begin{displaymath} w(X)_n:=\coprod_{T:ht(T)\le n}hom_{\omega Graph}(T_*,X) \end{displaymath} $\eta:id_{Psh(G)}\to w$ is induced by Yoneda: $X_n\mapsto hom_{\omega Graph}(n_*,X)$ \end{defn} \hypertarget{2_cellular_sets_and_their_geometric_realization}{}\subsection*{{2. Cellular sets and their geometric realization}}\label{2_cellular_sets_and_their_geometric_realization} \hypertarget{3_a_closed_model_structure_for_cellular_sets}{}\subsection*{{3. A closed model structure for cellular sets}}\label{3_a_closed_model_structure_for_cellular_sets} \hypertarget{4_homotopy_structure_for_weak_categories}{}\subsection*{{4. Homotopy structure for weak $\omega$-categories}}\label{4_homotopy_structure_for_weak_categories} \end{document}