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omegacat.tex

@@ -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 $k<n$, composition map $\iota(C)_n\underset{\iota(C)_k}{\times}\iota(C)_n \simeq \iota(C)_n \to \iota(C)_n$ as the identity.
+    \end{itemize}
+  It is immediate to see that $\iota(C)$ is indeed an $n$-category and the correspondance $ C \mapsto \iota(C)$ can canonically be made into a functor:
+  \[
+    \iota : (n\shortminus1)\Cat \to n\Cat.
+    \]
+ %   andy definition, for any $(n \shortminus 1)$-category, we have
+  %  \[
+  %  \tau(\iota(C))=C.
+  %  \]
+
+    We also define an $n$-category $\kappa(C)$ with
+    \begin{itemize}
+    \item[-]\tau(\kappa(C))=C,
+      \item[-]
+      \end{itemize}
+\end{paragr}
+\begin{lemma}
+  We have a sequence of adjunctions
+  \[
+\iota \dashv \tau \dashv \kappa.
+  \]
+  \end{lemma}
 \section{Generating cells}
 \begin{paragr}
   Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product