For any $n>0$, there is an obvious ``truncation'' functor

\[

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

\]

\end{paragr}

\begin{remark}

More concretely, the definitions of $\omega$-graph and morphism of $\omega$-graphs are the same as the definitions of $n$-graph and morphisms of $n$-graphs with $n$ finite but with the condition ``$\leq n$'' dropped everywhere. For example, an $\omega$-graph has an infinite sequence of cells $(C_k)_{k\in\mathbb{N}}$.

\remtt{Est-ce que je devrais plus détailler la définition de $\oo$-graphe ?}

\end{remark}

that simply forgets the $n$-cells. That is to say, for $C$ an $n$-graph, $\tau(C)$ is the $(n \shortminus1)$-graph with $\tau(C)_k=C_k$ for every $0\leq k \leq n\shortminus1$.

\end{paragr}

\begin{paragr}

Let $n \in\mathbb{N}$. An \emph{$n$-magma} consists of:

\begin{itemize}

...

...

@@ -153,32 +146,21 @@

\]

\end{itemize}

We denote by $\nMag$ the category of $n$-magmas and morphisms of $n$-magmas.

\end{paragr}

\begin{paragr}

For any $n>0$, there is an obvious ``truncation'' functor

For any $n>0$, there is an obvious ``truncation'' functor

\[

n\Mag\to(n\shortminus1)\Mag

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

\]

that simply forgets the $n$-cells. We define the category $\oo\Mag$ of $\omega$-magmas as the limit of the diagram

that simply forgets the $n$-cells. Moreover, the square

where the vertical arrows are the obvious forgetful functors, is commutative.

\end{paragr}

\begin{paragr}

Let $n \in\mathbb{N}$. An \emph{$n$-category}$C$ is an $n$-magma such that the following axioms are satisfied:

\begin{enumerate}

...

...

@@ -206,6 +188,140 @@

\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}

\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{tikzcd}

n\CellExt\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]&(n \shortminus1)\Cat\ar[d]\\

n\Grph\ar[r]&(n \shortminus1)\Grph.

\end{tikzcd}

\]

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.

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}

% \Sigma \ar[r,shift left,"s"] \ar[r,shift right,"t"']& C

% \end{tikzcd}

%\]

\end{paragr}

\begin{paragr}

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

\[

\begin{tikzcd}

n\PCat\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]&(n \shortminus1)\Cat\ar[d]\\

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. 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 commutative square

\[

\begin{tikzcd}

n\Cat\ar[r]\ar[d]&(n \shortminus1)\Cat\ar[d]\\

n\Mag\ar[r]&(n \shortminus1)\Mag

\end{tikzcd}

\]

induces a canonical functor

\[

V : n\Cat\to n\PCat,

\]

which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.

More concretely, the definitions of $\omega$-graph and morphism of $\omega$-graphs are the same as the definitions of $n$-graph and morphisms of $n$-graphs with $n$ finite but with the condition ``$\leq n$'' dropped everywhere. For example, an $\omega$-graph has an infinite sequence of cells $(C_k)_{k\in\mathbb{N}}$.

\remtt{Est-ce que je devrais plus détailler la définition de $\oo$-graphe ?}

\end{remark}

\begin{paragr}

For any $n>0$, there is an obvious ``truncation'' functor

\[

n\Mag\to(n\shortminus1)\Mag

\]

that simply forgets the $n$-cells. We define the category $\oo\Mag$ of $\omega$-magmas as the limit of the diagram

Moreover, for any $n\in\mathbb{N}$, there is a canonical forgetful functor

\[

n\Mag\to n\Grph

\]

such that when $n>0$ the square

\[

\begin{tikzcd}

n\Mag\ar[r]\ar[d]& n\Grph\ar[d]\\

(n \shortminus1)\Mag\ar[r]&(n \shortminus1)\Grph

\end{tikzcd}

\]

is commutative. By universal property, this yields a canonical forgetful functor

\[

\oo\Mag\to\oo\Grph.

\]

\end{paragr}

\begin{paragr}\label{paragr:defoocat}

Once again, for any $n>0$ there is a canonical ``truncation'' functor

\[

...

...

@@ -311,97 +427,3 @@

\begin{proof}

\todo{...}

\end{proof}

\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{tikzcd}

n\CellExt\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]&(n \shortminus1)\Cat\ar[d]\\

n\Grph\ar[r]&(n \shortminus1)\Grph.

\end{tikzcd}

\]

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.

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}

% \Sigma \ar[r,shift left,"s"] \ar[r,shift right,"t"']& C

% \end{tikzcd}

%\]

\end{paragr}

\begin{paragr}

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

\[

\begin{tikzcd}

n\PCat\ar[d]\ar[r]\ar[dr,phantom,"\lrcorner", very near start]&(n \shortminus1)\Cat\ar[d]\\

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. 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 commutative square

\[

\begin{tikzcd}

n\Cat\ar[r]\ar[d]&(n \shortminus1)\Cat\ar[d]\\

n\Mag\ar[r]&(n \shortminus1)\Mag

\end{tikzcd}

\]

induces a canonical functor

\[

V : n\Cat\to n\PCat,

\]

which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.