The ``truncation'' functors $\tau : n\Mag\to(n\shortminus1)\Mag$ and $\tau : n\Grph\to(n\shortminus1)\Grph$ also have left and right adjoints but we won't need them in the sequel.
\end{remark}
\section{Basis for $n$-categories}
\section{Bases for $n$-categories}
\begin{paragr}
Let $C$ be an $n$-category and $k\in\mathbb{N}$ with $k<n$. A \emph{$k$-prebasis} of $C$ is a subset
Let $C$ be an $n$-category and $k\in\mathbb{N}$ with $0\leq k \leq n$. A subset of $k$-cells
\[
\Sigma\subseteq C_k
\]
such that $C_k$ is the smallest subset
\end{paragr}
is \emph{saturated} if $k=0$ and $\Sigma= C_0$, or $k>0$ and:
\begin{itemize}
\item[-] for every $x \in C_{k\shortminus1}$, we have
\[
1_x \in\Sigma,
\]
\item[-] for all $x, y \in\Sigma$ that are $l$-composable with $l<k$, we have
\[
x\comp_ly \in\Sigma.
\]
\end{itemize}
It is immediate to see that any intersection of saturated subset of $k$-cells is again satured. For any arbitrary subset of $k$-cells $\Sigma\subseteq C_k$, we denote by $\overline{\Sigma}$ the smallest saturated subset of $k$-cells that contains $\Sigma$.
\end{paragr}
\begin{definition}
Let $C$ be an $n$-category and $k\in\mathbb{N}$ with $0\leq k \leq n$. We say that subset of $k$-cells $\Sigma\subseteq C_k$ is a \emph{$k$-prebasis} when
\[
\overline{\Sigma}=C_k.
\]
\end{definition}
\begin{definition}
\end{definition}
\section{Generating cells}
\begin{paragr}
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\item[-] a symbol $\hat{x}$ for each $x \in\Sigma$,
\item[-] a symbol $\hat{\comp_k}$ for each $k<n$,
\item[-] a symbol
\item[-] a symbol $\ii_x$ for each $x \in C_{n\shortminus1}$,
\item[-] a symbol of opening parenthesis $($,
\item[-] a symbol of closing parenthesis $)$.
\end{itemize}
\end{paragr}
We denote by $\W[E]$ the set of words on this alphabet (i.e. finite sequence of symbols). If $w$ and $w'$ are elements of $\mathcal{W}[E]$, we write $ww'$ for their concatenation.
The \emph{length} of a word $w$, denoted by $\mathcal{L}(w)$, is the number of symbols that appear in $w$.
\end{paragr}
\begin{paragr}
We now recursively define the set $\Sigma^{+}\subseteq\W[\Sigma]$ of \emph{well formed words} (or \emph{terms}) on this alphabet together with maps $\sigma,\tau : \Sigma^{+}\to C_{n-1}$ that satisfy the globular conditions:
\begin{itemize}
\item[-]$(\hat{x})\in\T[E]$ with $\sigma((\hat{x}))=\sigma(\alpha)$ and $t((\hat{x}))=\tau((\hat{x}))$ for each $x \in\Sigma$,
%\item[-] $(\ii_{x}) \in \T[E]$ with $s((\ii_x))=t((\ii_x))=x$ for each $x \in C_n$,
%\item[-] $ (v \ast_n w) \in \T[E]$ with $s((v \ast_n w))=s(w)$ and $t((v \ast_n w))=t(v)$ for $v,w \in \T[E]$ such that $s(v)=t(w)$,
% \item[-] $(v \ast_k w) \in \T[E]$ with \[s((v \ast_k w)) = s(v) \ast_k s(w)\] and \[t((v \ast_k w))=t(v)\ast_k t(w)\] for $v, w \in \T[E]$ and $0 \leq k < n$, such that $s^k(s(v))=t^k(t(w))$.
\end{itemize}
%As usual, if $w \in \T[E]$ we often write $w : x \to y$ to say that $s^n(w)=x$ and $t^n(w)=y$.
We define $s^k , t^k: \T[E]\to C_k$ as iterated source and target (with $s^n=s$ and $t^n=t$ for consistency). We say that two well formed words $v$ and $w$ are \emph{parallel} if
\[s(v)=s(w)\text{ and }t(v)=t(w)\]
and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\[s^k(v)=t^k(w).\]
\end{paragr}
\section{$\oo$-categories}
\begin{paragr}
For any $n>0$, there is an obvious ``truncation'' functor
