Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\begin{equation}\label{squarecellext}
\begin{tikzcd}
n\CellExt\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat\ar[d]\\
n\Grph\ar[r,"\tau"]& (n \shortminus 1)\Grph,
\end{tikzcd}
\end{equation}
where the right vertical arrow is the obvious forgetful functor.
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
\end{definition}
\section{Generating cells}
\begin{paragr}
A \emph{cellular extension} of an $n$-magma $M$ in the data of a quadruple $(\Sigma,M,s,t)$ where $\Sigma$ is a set, $M$ is a $n$-magma, $s$ and $t$ are maps
to denote an $n$-cellular extension $(\Sigma,C,s,t)$. \remtt{Est-ce que je garde cette notation ?}
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
Once again, we will use the notation $\tau$ for the functor
\[
\begin{aligned}
\tau : n\CellExt&\to(n\shortminus1)\Cat\\
(\Sigma,C,s,t)&\mapsto C
\end{aligned}
\]
which is simply the top horizontal arrow of square \eqref{squarecellext}.
satisfy the globular identities. We will often use the abusive notation $(\Sigma,M)$ instead of $(\Sigma,M,s,t)$.
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\begin{equation}\label{squareprecat}
\begin{tikzcd}
n\PCat\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat\ar[d]\\
n\Mag\ar[r,"\tau"]& (n \shortminus 1)\Mag.
\end{tikzcd}
\end{equation}
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 top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation
We denote by $\W[\Sigma]$ the set of words on this alphabet (i.e. finite sequence of symbols). If $w$ and $w'$ are elements of $\mathcal{W}[\Sigma]$, 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 $w$.
We now recursively define the set $\Sigma^{+}\subseteq\W[\Sigma]$ of \emph{well formed words} on this alphabet together with maps $s,t : \Sigma^{+}\toC_{n-1}$:
We now recursively define the set $\Sigma^{+}\subseteq\W[\Sigma]$ of \emph{well formed words} on this alphabet together with maps $s,t : \Sigma^{+}\toM_{n-1}$:
\begin{itemize}
\item[-] for every $x \in\Sigma$, we have $(\hat{x})\in\Sigma^{+}$ with
\[s((\hat{x}))=s(x)\text{ and }t((\hat{x}))=t(x),\]
\item[-] for every $x \in C_n$, we have $(\ii_{x})\in\Sigma^{+}$ with
\[s((\ii_x))=t((\ii_x))=x,\]
\item[-] for every $x \in\Sigma$, we have $\hat{x}\in\Sigma^{+}$ with
\[s(\hat{x})=s(x)\text{ and }t(\hat{x})=t(x),\]
\item[-] for every $x \in C_n$, we have $\ii_{x}\in\Sigma^{+}$ with
\[s(\ii_x)=t(\ii_x)=x,\]
\item[-] for all $v,w \in\Sigma^{+}$ such that $s(v)=t(w)$, we have $(v \fcomp_n w)\in\Sigma^{+}$ with \[s((v \fcomp_n w))=s(w)\text{ and }t((v \fcomp_n w))=t(v),\]
\item[-] for all $v, w \in\Sigma^{+}$ and $0\leq k < n\shortminus1$, such that $s_k(s(v))=t_k(t(w))$, we have $(v \fcomp_k w)\in\Sigma^{+}$ with \[s((v \hat{\comp_k} w))= s(v)\comp_k s(w)\] and \[t((v \hat{\comp_k} w))=t(v)\comp_k t(w).\]
\end{itemize}
We define $s_k , t_k: \Sigma^{+}\toC_k$ 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
We define $s_k , t_k: \Sigma^{+}\toM_k$ 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 $k< n$ if
\[s_k(v)=t_k(w).\]
Let $E'=(\Sigma',C',s',t')$ be another $n$-cellular extension and $(\varphi,f) : E \to E'$ a morphism of $n$-cellular extensions. We recursively define a map $f^+ : \Sigma^+\to\Sigma^+$ with
\begin{itemize}
\item[-]
\end{itemize}
%Let $E'=(\Sigma',C',s',t')$ be another $n$-cellular extension and $(\varphi,f) : E \to E'$ a morphism of $n$-cellular extensions. We recursively define a map $f^+ : \Sigma^+ \to \Sigma^+$ with
%\begin{itemize}
% \item[-]
%\end{itemize}
\end{paragr}
\begin{paragr}
Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with
\begin{itemize}
\item[-]$\tau(W_!(E))=C$,
\item[-]$W_!(E)_n=\Sigma^{+}$,
\item[-] source and target maps $\Sigma^+\to C_{n-1}$ as defined in the previous paragraph,
\item[-] for every $x \in C_{n-1}$,
\[1_x :=(\ii_x)\]
\item[-] for every $v,w \in\Sigma^+$ that are $k$-composable for a $k<n$,
Let $C$ be an $n$-category, $k \in\mathbb{N}$ with $0<k \leq n$ and $\Sigma\subseteq C_k$ a subset of the $k$-cells of $C$ with $k\leq n$. It defines an cellular extension $(\Sigma,\tau_{k-1}(C),\sigma,\tau)$ of $\tau_{k-1}(C)$, where $s$ and $t$ are simply the restriction of the source and target maps $C_k \to C_{k-1}$. The canonical inclusion
\[
v\comp_kw :=(v \fcomp_k w).
\Sigma\hookrightarrow C_k
\]
\end{itemize}
It is straightforward to check that this defines an $n$-precategory. Let $(\varphi,f) : E \to E'$ be a morphism of $n$-cellular extensions. We define a morphism of $n$-precategories $W_!(\varphi,f) : W_!(E)\to W_!(E')$ with
is recursively extended to a map $\rho : \Sigma^{+}\to C_k$ in the following way:
\begin{enumerate}[label=-]
\item$\rho(\hat{x})=x$ for every $x \in\Sigma$,
\item$\rho((v\fcomp_kw))=\rho(v)\comp_k\rho(w)$.
\end{enumerate}
We call $\rho$ the \emph{evaluation map}. Intuitively, this maps ``evaluates'' the formal expressions constructed out of the cells in $\Sigma$ into actual $k$-cells of $\Sigma$.
\end{paragr}
\begin{definition}
Let $C$ be an $n$-category and $k\leq n$. A subset $\Sigma\subseteq C_k$ of $k$-cells \emph{generates by composition} if the evaluation map
\[
\rho : \Sigma^+\to C_k
\]
is surjective.
\end{definition}
\begin{definition}
Let $M$ be an $n$-magma and $k \leq n$. A \emph{congruence} on the $k$-cells of $M$ is a binary relation $\R$ on $M_k$ such that
\begin{enumerate}[label=-]
\item$\R$ is an equivalence relation,
\item if $x\R y$ then $x$ and $y$ are parallel,
\item for $x, x', y, y' \in M_k$ such that $x$ and $x'$ are $l$-composable, $y$ and $y'$ are $l$-composable if $x\R y$ and $x'\R y'$, then $(x\ast_l y )\R(x'\ast_l y')$.
\end{enumerate}
\end{definition}
\begin{lemma}
Let $M$ be an $n$-magma and $(\R_i)_{i \in I}$ a family of congruence on the $k$-cells of $M$. If $I$ is non-empty then $\cap_{i \in I}\R_i$ is a congruence on the $k$-cells of $M$.
\end{lemma}
\
\section{$\oo$-categories}
\begin{paragr}
For any $n>0$, there is an obvious ``truncation'' functor
@@ -192,3 +192,147 @@ and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\end{tikzcd}
\]
is an $n$-graph and
\section{Generating cells}
\begin{paragr}
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\begin{equation}\label{squarecellext}
\begin{tikzcd}
n\CellExt\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat\ar[d]\\
n\Grph\ar[r,"\tau"]& (n \shortminus 1)\Grph,
\end{tikzcd}
\end{equation}
where the right vertical arrow is the obvious forgetful functor.
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
Intuitively, a $n$-cellular extension is a $(n\shortminus1)$-category with extra $n$-cells that make it a $n$-graph.
We will sometimes write
\[
\begin{tikzcd}
C &\ar[l,shift right,"s"']\ar[l,shift left,"t"]\Sigma
\end{tikzcd}
\]
to denote an $n$-cellular extension $(\Sigma,C,s,t)$. \remtt{Est-ce que je garde cette notation ?}
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
Once again, we will use the notation $\tau$ for the functor
\[
\begin{aligned}
\tau : n\CellExt&\to(n\shortminus1)\Cat\\
(\Sigma,C,s,t)&\mapsto C
\end{aligned}
\]
which is simply the top horizontal arrow of square \eqref{squarecellext}.
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\begin{equation}\label{squareprecat}
\begin{tikzcd}
n\PCat\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat\ar[d]\\
n\Mag\ar[r,"\tau"]& (n \shortminus 1)\Mag.
\end{tikzcd}
\end{equation}
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 top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation
We will now explicitely construct a left adjoint of $U$. In order to do that, we will successively construct left adjoints of $W$ and $V$.
\end{paragr}
\begin{paragr}
Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with
\begin{itemize}
\item[-]$\tau(W_!(E))=C$,
\item[-]$W_!(E)_n=\Sigma^{+}$,
\item[-] source and target maps $\Sigma^+\to C_{n-1}$ as defined in the previous paragraph,
\item[-] for every $x \in C_{n-1}$,
\[1_x :=(\ii_x)\]
\item[-] for every $v,w \in\Sigma^+$ that are $k$-composable for a $k<n$,
\[
v\comp_kw :=(v \fcomp_k w).
\]
\end{itemize}
It is straightforward to check that this defines an $n$-precategory. Let $(\varphi,f) : E \to E'$ be a morphism of $n$-cellular extensions. We define a morphism of $n$-precategories $W_!(\varphi,f) : W_!(E)\to W_!(E')$ with
\end{paragr}
For $u : A \to B$ in $\CCat$, let
\[
u^* : \C(A)\to\C(B)
\]
be the functor induced by post-composition. For $\begin{tikzcd}\sD(B)\ar[r,bend left,"u^*",""{name=U,below}]\ar[r,bend right,"v^*"',""{name=D,above}]&\sD(A)\ar[from=U,to=D,Rightarrow,"\alpha^*"]\end{tikzcd}$
Note that we have a canonical isomorphism
\[
\C(e)\simeq\C
\]
and for any small category $A$, the functor
\[
p_A^* : \C(e)\to\C(A)
\]
is canonically isomorphic with the diagonal functor $\Delta : \C\to\C(A)$ that sends an object $X$ of $\C$ to the constant diagram $A \to\C$ with value $X$.
