\remtt{Le lemme suivant est un genre d'Eilenberg-Zilber pour les oo-catégories.}

\begin{lemma}\label{EZoocat}

Let $C$ be an $\oo$-category and $n \in\mathbb{N}$. For any $n$-cell $x$ of $C$, there exist a unique non-degenerate $k$-cell $x'$ with $k\leq n$ such that

\[

\1^n_{x'}=x.

\]

\end{lemma}

\begin{proof}

\todo{Évident. À écrire ?}

\end{proof}

\begin{paragr}

For any $n>0$, let us denote $\tau$ the canonical truncation functor

\[

\tau : n\Cat\to(n \shortminus1)\Cat

\]

from Paragraph \ref{paragr:defoocat} and

from Paragraph \ref{paragr:defoocat}.

Let $C$ be a $(n\shortminus1)$-category. We define an $n$-magma $\iota(C)$ with

\begin{itemize}

\item[-]$\tau(\iota(C))=C$,

\item[-]$C'_{n}=C_{n-1}$,

\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 in fact an $n$-category and the correspondance $ C \mapsto\iota(C)$ can canonically made into a functor:

that we call the \emph{canonical filtration of $C$} and the sequence of arrows

\[

(\eta_n : \sk_n(C)\to C)_{n \in\mathbb{N}}

\]

where $\eta_n$ is the unit of the adjunction $\iota^n \dashv\tau_n$, defines a cocone on the previous diagram.

\end{paragr}

\begin{lemma}

Every $\oo$-category is the colimit of its canonical filtration.

\end{lemma}

\begin{proof}

\todo{...}

\end{proof}

\section{Generating cells}

\begin{paragr}

Let $n>0$, we define the category $n\GCat$ as the following fibred product

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

\[

\begin{tikzcd}

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

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.

\end{tikzcd}

\]

More concretely, an object of $n\GCat$ consists of 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

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

\[

s,t : \Sigma\to C_n

\]

...

...

@@ -263,13 +331,23 @@

\end{tikzcd}

\]

satisfy the globular identities.

A morphism of $n\GCat$ 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

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

%From now on, we will denote such an object of $n\GCat$ by

%\[

%\begin{tikzcd}

...

...

@@ -285,18 +363,27 @@

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.

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 $n\Mag$.