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\begin{document}

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\section*{globular set}

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

\hypertarget{graph_theory}{}\paragraph*{{Graph theory}}\label{graph_theory}

[[!include graph theory - contents]]

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

[[!include higher category theory - contents]]

\hypertarget{globular_sets}{}\section*{{Globular sets}}\label{globular_sets}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{basic_definition}{Basic definition}\dotfill \pageref*{basic_definition} \linebreak
\noindent\hyperlink{ReflexiveGlobularSets}{Reflexive globular sets}\dotfill \pageref*{ReflexiveGlobularSets} \linebreak
\noindent\hyperlink{globular_sets_2}{$n$-globular sets}\dotfill \pageref*{globular_sets_2} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{grothendieck_homotopy_theory}{Grothendieck homotopy theory}\dotfill \pageref*{grothendieck_homotopy_theory} \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}

Globular sets are to [[simplicial sets]] as [[globes]] are to simplices.

They are one of the major [[geometric shapes for higher structures]]: if they satisfy a globular [[Segal condition]] then they are equivalent to [[strict ∞-categories]].

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

\hypertarget{basic_definition}{}\subsubsection*{{Basic definition}}\label{basic_definition}

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{[[globe category]]} $\mathbb{G}$ is the [[category]] whose [[objects]] are the [[natural numbers]], denoted here $[n] \in \mathbb{N}$ (N.B. not to be confused with ordinals in any structural sense) and whose morphisms are [[generators and relations|generated]] from

\begin{displaymath}
\sigma_n : [n] \to [n+1]
\end{displaymath}
\begin{displaymath}
\tau_n : [n] \to [n+1]
\end{displaymath}
for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)

\begin{displaymath}
\sigma\circ \sigma = \tau \circ \sigma
\end{displaymath}
\begin{displaymath}
\sigma\circ \tau = \tau \circ \tau
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \emph{globular set}, also called an \emph{$\omega$-[[graph]]}, is a [[presheaf]] on $\mathbb{G}$. The [[category]] of globular sets is the [[category of presheaves]]

\begin{displaymath}
gSet \coloneqq PSh(\mathbb{G})
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This means that a globular set $X \in gSet$ is given by a collection of sets $\{X_n\}_{n \in \mathbb{N}}$ (the \textbf{set of $n$-[[globes]]}) equipped with [[functions]]

\begin{displaymath}
\{s_n,t_n \colon X_{n+1} \to X_n\}_{n \in \mathbb{N}}
\end{displaymath}
called the \textbf{$n$-source} and \textbf{$n$-target} maps (or similar), such that the \textbf{globular identities} hold: for all $n \in \mathbb{N}$

\begin{itemize}%
\item $s_n \circ s_{n+1} = s_n \circ t_{n+1}$


\item $t_n \circ s_{n+1} = t_n \circ t_{n+1}$.



\end{itemize}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The globular identities ensure that two sequences of boundary maps

\begin{displaymath}
f_n \circ \cdots \circ f_{n+m-1} \circ f_{n+m} :
  S_{n+m+1} \to S_n
\end{displaymath}
with $n,m \in \mathbb{N}$ and for $f_k, \in \{s_k, t_k\}$ are equal if and only if their last term $f_n$ coincides; for all $n,m \in \mathbb{N}$ we have

\begin{displaymath}
s_n \cdots s_{n+1} \circ \cdots \circ
    s_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n
    =
    \mathrm{Id}
\end{displaymath}
\begin{displaymath}
t_n \cdots t_{n+1} \circ \cdots \circ
    t_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n
    =
    \mathrm{Id}
    \,.
\end{displaymath}
For $S$ a globular set we may therefore write unambiguously

\begin{displaymath}
s_n, t_n : S_{n+m+1} \to S_n
\end{displaymath}
\begin{displaymath}
i_n : S_n \to S_{n+m+1}
\end{displaymath}
with $i_n, s_n, t_m$ the sequence of $m$ consecutive identity-assigning, source or target maps, respectively.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The presheaf definition can understood from the point of view of [[space and quantity]]: a \emph{globular set} is a space characterized by the fact that and how it may be \emph{probed} by mapping standard globes into it: the set $S_n$ assigned by a globular set to the standard $n$-globe $[n]$ is the set of $n$-globes in this space, hence the way of mapping a standard $n$-globe into this spaces.

\end{remark}
More generally:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{globular object} $X$ [[internalization|in]] a [[category]] $\mathcal{C}$ is a [[functor]] $X : \mathbb{G}^{\mathrm{op}} \to \mathcal{C}$.

\end{defn}
\hypertarget{ReflexiveGlobularSets}{}\subsubsection*{{Reflexive globular sets}}\label{ReflexiveGlobularSets}

If to the globe category we add additional generating morphisms

\begin{displaymath}
\iota_n : [n+1] \to [n]
\end{displaymath}
satisfying the relations

\begin{displaymath}
\iota \circ \sigma = \mathrm{Id}
\end{displaymath}
\begin{displaymath}
\iota \circ \tau = \mathrm{Id}
\end{displaymath}
we obtain the \textbf{reflexive globe category}, a presheaf on which is a \textbf{reflexive globular set}. In this case the morphism

\begin{displaymath}
i_n := S(\iota_n) : S_{n} \to S_{n+1}
\end{displaymath}
is called the $n$th \textbf{identity assigning map}; it satisfies the globular identities:

\begin{displaymath}
s \circ i = \mathrm{Id}
\end{displaymath}
\begin{displaymath}
t \circ i = \mathrm{Id}
\end{displaymath}
\hypertarget{globular_sets_2}{}\subsubsection*{{$n$-globular sets}}\label{globular_sets_2}

A presheaf on the full subcategory of the globe category containing only the integers $[0]$ through $[n]$ is called an \textbf{$n$-globular set} or an \textbf{$n$-graph}. An $n$-globular set may be identified with an $\infty$-globular set which is empty above dimension $n$.

Note that a $1$-globular set is just a [[directed graph]], and a $0$-globular set is just a [[set]].

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

\begin{itemize}%
\item Any [[strict 2-category]] or [[bicategory]] has an underlying 2-globular set. Likewise, any [[tricategory]] has an underlying 3-globular set. Globular sets can be used as underlying data for [[n-categories]] as well; see for instance [[Batanin ∞-category]].


\item A \emph{[[strict omega-category]]} is a globular set $C$ equipped in each degree with the structure of a [[category]] such that for every pair $k_1 \lt k_2 \in \mathbb{N}$ the induced structure on the 2-graph $C_{k_2} \stackrel{\to}{\to} C_{k_1} \stackrel{\to}{\to} C_0$ is that of a [[strict 2-category]].


\item The \textbf{globular $n$-globe} $G_n$ is the globular set represented by $n$, i.e. $G_n(-) := Hom_G(-,n)$.



\end{itemize}
\hypertarget{grothendieck_homotopy_theory}{}\subsection*{{Grothendieck homotopy theory}}\label{grothendieck_homotopy_theory}

The category of [[globes]] is not a [[weak test category]] according to Scholium 8.4.14 in Cisinski \ref{PMTH}.

However, if we construct the free [[strict monoidal category]] on the category of [[globes]], while ensuring that the terminal object becomes the monoidal unit, then the resulting category of polyglobes is a [[test category]].

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

\begin{itemize}%
\item [[graph]]


\item [[directed graph]]

\begin{itemize}%
\item \textbf{directed $n$-graph}

\end{itemize}

\item [[ribbon graph]]


\item [[Theta category]],


\item [[globular theory]], [[globular operad]]


\item Also related is the notion of [[computad]], which is similar to a globular set in some ways, but allows ``formal composites'' of $n$-cells to appear in the sources and targets of $(n+1)$-cells.


\item [[simplicial object]]

\begin{itemize}%
\item [[simplicial set]]


\item [[simplicial object in an (∞,1)-category]]



\end{itemize}

\item [[semi-simplicial object]]

\begin{itemize}%
\item [[semisimplicial set]]

\end{itemize}


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

The definition is reviewed around def. 1.4.5, p. 49 of

\begin{itemize}%
\item [[Tom Leinster]]: \emph{Higher operads, higher categories} (\href{http://arxiv.org/abs/math/0305049}{arXiv:math/0305049})

\end{itemize}
See also

\begin{itemize}%
\item [[Sjoerd Crans]], \emph{On combinatorial models for higher dimensional homotopies} (\href{https://web.archive.org/web/20130514164856/http://home.tiscali.nl/secrans/papers/comb.html}{web})


\item [[R. Street]], \emph{The petit topos of globular sets} , JPAA \textbf{154} (2000) pp.299-315.



\end{itemize}
$\backslash$bibitem\{PMTH\} [[Denis-Charles Cisinski]], \emph{Les préfaisceaux comme modèles des types d’homotopie}, Asterisque.

The definition of globular set, without using that term, is in 2.2 and 2.3 of the following paper:

\begin{itemize}%
\item [[Ronnie Brown]] and Philip J. Higgins, ``The equivalence of $\infty$-groupoids and crossed complexes'', Cah. Top. G'eom. Diff. 22 (1981) 371-386.

\end{itemize}
The following paper constructs from the cubical case a strict globular $\omega$-groupoid of a filtered space:

\begin{itemize}%
\item [[Ronnie Brown]] ``A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space'', Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.

\end{itemize}
[[!redirects globular set]] [[!redirects globular sets]] [[!redirects reflexive globular set]] [[!redirects reflexive globular sets]] [[!redirects n-globular set]] [[!redirects n-globular sets]] [[!redirects reflexive n-globular set]] [[!redirects reflexive n-globular sets]] [[!redirects omega-graph]] [[!redirects omega-graphs]] [[!redirects ∞-graph]] [[!redirects ∞-graphs]] [[!redirects infinity-graph]] [[!redirects infinity-graphs]] [[!redirects ∞-graph]] [[!redirects ∞-graphs]] [[!redirects n-graph]] [[!redirects n-graphs]] [[!redirects omega-quiver]] [[!redirects omega-quivers]] [[!redirects ∞-quiver]] [[!redirects ∞-quivers]] [[!redirects infinity-quiver]] [[!redirects infinity-quivers]] [[!redirects ∞-quiver]] [[!redirects ∞-quivers]] [[!redirects n-quiver]] [[!redirects n-quivers]] [[!redirects directed omega-graph]] [[!redirects directed omega-graphs]] [[!redirects directed ∞-graph]] [[!redirects directed ∞-graphs]] [[!redirects directed infinity-graph]] [[!redirects directed infinity-graphs]] [[!redirects directed ∞-graph]] [[!redirects directed ∞-graphs]] [[!redirects directed n-graph]] [[!redirects directed n-graphs]] [[!redirects omega-directed graph]] [[!redirects omega-directed graphs]] [[!redirects ∞-directed graph]] [[!redirects ∞-directed graphs]] [[!redirects infinity-directed graph]] [[!redirects infinity-directed graphs]] [[!redirects ∞-directed graph]] [[!redirects ∞-directed graphs]] [[!redirects n-directed graph]] [[!redirects n-directed graphs]] [[!redirects reflexive omega-graph]] [[!redirects reflexive omega-graphs]] [[!redirects reflexive ∞-graph]] [[!redirects reflexive ∞-graphs]] [[!redirects reflexive infinity-graph]] [[!redirects reflexive infinity-graphs]] [[!redirects reflexive ∞-graph]] [[!redirects reflexive ∞-graphs]] [[!redirects reflexive n-graph]] [[!redirects reflexive n-graphs]] [[!redirects reflexive omega-quiver]] [[!redirects reflexive omega-quivers]] [[!redirects reflexive ∞-quiver]] [[!redirects reflexive ∞-quivers]] [[!redirects reflexive infinity-quiver]] [[!redirects reflexive infinity-quivers]] [[!redirects reflexive ∞-quiver]] [[!redirects reflexive ∞-quivers]] [[!redirects reflexive n-quiver]] [[!redirects reflexive n-quivers]] [[!redirects reflexive directed omega-graph]] [[!redirects reflexive directed omega-graphs]] [[!redirects reflexive directed ∞-graph]] [[!redirects reflexive directed ∞-graphs]] [[!redirects reflexive directed infinity-graph]] [[!redirects reflexive directed infinity-graphs]] [[!redirects reflexive directed ∞-graph]] [[!redirects reflexive directed ∞-graphs]] [[!redirects reflexive directed n-graph]] [[!redirects reflexive directed n-graphs]] [[!redirects reflexive omega-directed graph]] [[!redirects reflexive omega-directed graphs]] [[!redirects reflexive ∞-directed graph]] [[!redirects reflexive ∞-directed graphs]] [[!redirects reflexive infinity-directed graph]] [[!redirects reflexive infinity-directed graphs]] [[!redirects reflexive ∞-directed graph]] [[!redirects reflexive ∞-directed graphs]] [[!redirects reflexive n-directed graph]] [[!redirects reflexive n-directed graphs]]



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