\section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories}

\begin{paragr}\label{pargr:defngraph}

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@@ -316,8 +316,8 @@

Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product

\[

\begin{tikzcd}

n\CellExt\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]&n\Grph\ar[d]\\

(n \shortminus1)\Cat\ar[r]&(n \shortminus1)\Grph.

n\CellExt\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]&(n \shortminus1)\Cat\ar[d]\\

n\Grph\ar[r]&(n \shortminus1)\Grph.

\end{tikzcd}

\]

More concretely, an $n$-cellular extension can be encoded in the data of a quadruple $(\Sigma,C,s,t)$ where $\Sigma$ is a set, $C$ is a $(n\shortminus1)$-category, $s$ and $t$ are maps

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@@ -332,7 +332,7 @@

\]

satisfy the globular identities.

Intuitively, a $n$-cellular extension is a $(n\shortminus1)$-category with some extra $n$-cells that makes it a $n$-graph.

Intuitively, a $n$-cellular extension is a $(n\shortminus1)$-category with extra $n$-cells that make it a $n$-graph.

A morphism of $n$-cellular extensions from $(\Sigma,C,s,t)$ to $(\Sigma',C',s',t')$ consists of a pair $(\varphi,f)$ where $f : C \to C'$ is a morphism of $(n\shortminus1)\Cat$ and $\varphi : \Sigma\to\Sigma'$ is a map such that the squares

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@@ -363,7 +363,7 @@

n\Mag\ar[r]&(n \shortminus1)\Mag.

\end{tikzcd}

\]

More concretely, objects of $n\PCat$ can be seen as $n$-magmas such that if we forget their $n$-cells then they satisfy the axioms of $(n\shortminus1)$-category. The left vertical arrow of the previous square is easily seen to be full, hence a morphism of $n$-precategories is just a morphism of underlying $n$-magmas. We will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$.

More concretely, objects of $n\PCat$ can be seen as $n$-magmas such that if we forget their $n$-cells then they satisfy the axioms of $(n\shortminus1)$-category. The left vertical arrow of the previous square is easily seen to be full, and we will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$.

The commutative square

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@@ -377,13 +377,31 @@

\[

V : n\Cat\to n\PCat,

\]

which is also easily seen to be full.

which is also easily seen to be full.%and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.