Commit e0cb1427 by Leonard Guetta

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parent 0c97b4a7
 ... ... @@ -131,7 +131,7 @@ $\1^{k}_x := x.$ A cell is \emph{degenerate} if it is a unit on a strictly lower dimensional cell. A cell is \emph{degenerate} or \emph{trivial} if it is a unit on a strictly lower dimensional cell. Let $C$ and $C'$ be $n$-magmas. A \emph{morphism of $n$-magmas} $f : C \to C'$ is a morphism of $n$-graphs that is compatible with compositions and units. This means: ... ... @@ -188,6 +188,48 @@ \end{enumerate} We will use the same letter to denote an $n$-category and its underlying $n$-magma. Let $C$ and $C'$ be $n$-categories, a \emph{morphism of $n$-categories} (or $n$-functor) $f : C \to C'$ is simply a morphism of $n$-magmas. We denote by $n\Cat$ the category of $n$-categories and morphisms of $n$-categories. \end{paragr} \begin{paragr} Let $n>0$. As in the case of $n$-graphs and $n$-magmas, there is an obvious truncation'' functor $\tau : n\Cat \to (n\shortminus 1)\Cat$ that simply forgets the $n$-cells and the square $\begin{tikzcd} n\Cat \ar[r,"\tau"]\ar[d] & (n\shortminus 1)\Cat\ar[d]\\ n\Mag \ar[r,"\tau"] & (n\shortminus 1)\Mag \end{tikzcd}$ where the vertical arrows are the obvious forgetul functors, is commutative. Now let $C$ be an $(n\shortminus 1)$-category. We define an $n$-category $\iota(C)$ with \begin{itemize} \item[-] $\tau(\iota(C))=C$, \item[-] $\iota(C)_{n}=C_{n-1}$, \item[-] source and targets maps $\iota(C)_{n} \to \iota(C)_{n-1}$ as the identity, \item[-] unit map $\iota(C)_{n-1} \to \iota(C)_n$ as the identity, \item[-] for every $k0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product ... ...
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