We will use the same letter to denote an $\oo$-category and its underlying $\oo$\nbd-magma. A \emph{morphism of $\oo$-categories} (or \emph{$\oo$-functor}), $f : X \to Y$, is simply a morphism of the underlying $\oo$\nbd-magmas. We denote by $\oo\Cat$ the category of $\oo$-categories and morphisms of $\oo$-categories. This category is clearly locally presentable.

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@@ -739,11 +739,11 @@ The following proposition is the key result of this section. It is slightly less

Concretely, Proposition \ref{prop:freeonpolygraph} gives us a recipe to construct free $\oo$-categories. It suffices to give a formal list of generating cells of the form:

where for a generating $k$-cell $x$ with $k>0$, $\sigma(x)$ and $\tau(x)$ are $(k-1)$-cells of the free $(k-1)$-category recursively generated by the generating cells of dimension strictly lower than $k$.

where for a generating $k$-cell $x$ with $k>0$, $\sigma(x)$ and $\tau(x)$ are parallel $(k-1)$-cells of the free $(k-1)$-category recursively generated by the generating cells of dimension strictly lower than $k$.

\end{paragr}

\begin{example}

The data of $1$-cellular extension $\E$ is nothing but the data of a graph $G$ (or $1$-graph in the terminology of of \ref{paragr:defncat}), and it is not hard to see that, in that case, $\E^*$ is nothing but the free category on $G$. That is to say, the category whose objects are those of $G$ and whose arrows are strings of composable arrows of $G$; the composition being given by concatenation of strings. Hence, from Proposition \ref{prop:freeonpolygraph}, a ($1$-)category is free in the sense of Definition \ref{def:freeoocat} if and only if it is (isomorphic to) a free category on a graph.

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@@ -771,3 +771,130 @@ The following proposition is the key result of this section. It is slightly less

\end{example}

\section{Cells of free $\oo$-categories as words}

In this section, we undertake to give a more explicit construction of the $(n+1)$\nbd-category $\E^*$ generated by an $n$-cellular cellular extension $\E=(C,\Sigma,\sigma,\tau)$. By definition of $\E^*$, this amounts to give an explicit description of a particular type of colimit in $\oo\Cat$. Note also that since $\tau_{\leq n}(\E^*)=C$, all we need to do is to describe the $(n+1)$-cells of $\E^*$. This will take place in two steps: first we construct what ought to be called the \emph{free $(n+1)$-magma generated by $\E$}, for which the $(n+1)$-cells are really easy to describe, and then we quotient out these cells as to obtain an $(n+1)$-category, which will be $\E^*$.

Recall that an $n$-category is a particular case of $n$-magma.

\begin{paragr}\label{paragr:defwords}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. We denote by $\W[\E]$ the set of finite words of the alphabet that has:

\begin{itemize}[label=-]

\item a symbol $\cc_{\alpha}$ for each $\alpha\in\Sigma$,

\item a symbol $\ii_{x}$ for each $x \in C_n$,

\item a symbol $\fcomp_k$ for each $0\leq k \leq n$,

\item a symbol of opening parenthesis $($,

\item a symbol of closing parenthesis $)$.

\end{itemize}

If $w$ and $w'$ are two elements of $\W[\E]$, we write $ww'$ for their concatenation. We now define the subset $\T[\E]\subseteq\W[\E]$ of \emph{well formed words} (or \emph{terms}) on the previous alphabet together with two maps $\src, \trgt : \T[E]\to C_n$ in the following recursive way:

\begin{itemize}[label=-]

\item for every $\alpha\in\Sigma$, the word $(\cc_{\alpha})$ is well formed and we have

\[

\src((\cc_{\alpha}))=\sigma(\alpha)\text{ and }\trgt((\cc_{\alpha}))=\tau(\alpha),

\]

\item for every $x \in C_n$, the word $(\ii_{x})$ is well formed and we have

\[

\src((\ii_{x}))=\trgt((\ii_{x}))=x,

\]

\item if $w$ and $w'$ are well formed words such that $\src(w)=\trgt(w')$, then the word $(w\fcomp_n w')$ is well formed and we have

\[

\src((w\fcomp_n w'))=s(w')\text{ and }\trgt((w\fcomp_n w'))=t(w),

\]

\item if $w$ and $w'$ are well formed words such that $\src_k(\src(w))=\trgt_k(\trgt(w))$ for some $0\leq k < n$, then the word $(w\fcomp_k w')$ is well formed and we have

\[

\src((w \fcomp_k w'))=\src(w)\comp_k s(w')\text{ and }\trgt((w \fcomp_k w'))=\trgt(w)\comp_k \trgt(w').

\]

\end{itemize}

As usual, for $0\leq k<n$, we define $\src_k,\trgt_k : \T[\E]\to C_k$ to be respectively the iterated source and target (and we set $\src_n=\src$ and $\trgt_n=\trgt$ for consistency).

\end{paragr}

\begin{paragr}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. Let $\E^{+}$ denote the $(n+1)$-magma defined in the following fashion:

\begin{itemize}[label=-]

\item for every $0\leq k \leq n$, we have $(\E^+)_k:=C_k$; the source, target, compositions of $k$-cells for $0 < k \leq n$ and units on $k$-cells for $0\leq k <n$ are the ones of $C$,

\item$(\E^{+})_{n+1}=\T[\E]$,

\item the source and target maps $\src,\trgt : (\E^{+})_{n+1}\to(\E^{+})_{n}$ are the ones defined in \ref{paragr:defwords},

\item for every $n$-cell $x$, the unit on $x$ is given by the word

\[

(\ii_x),

\]

\item for $0\leq k \leq n$, the $k$-composition of two $(n+1)$-cells $w$ and $w'$ such that $\src_k(w)=\trgt_k(w')$ is given by the word

\[

(w\fcomp_k w').

\]

\end{itemize}

Hence, by definition, $\E^+$ satisfy all the axioms of $\oo$-categories up to dimension $n$ only. On the other hand, the $(n+1)$-cells of $\E^+$ make it as far as possible as being a $(n+1)$-category as \emph{none} of the axioms of $\oo$-categories are satisfied for cells of dimension $n+1$.

\end{paragr}

%We would like now to quotient the set of $(n+1)$-cells of $\E^+$ as to make it an $(n+1)$-category. In order to do that, we introduce the notion of congruence.

In the following definition, we consider that a binary relation $\R$ on a set $E$ is nothing but a subset of $E \times E$, and we write $x \;\R\; x'$ to say $(x,x')\in\R$.

\begin{definition}\label{def:congruence}

Let $n\geq1$. A \emph{congruence} on an $n$-magma $X$ is a binary relation $\R$ on the set of $n$-cells $X_n$ such that:

\begin{enumerate}[label=(\alph*)]

\item$\R$ is an equivalence relation,

\item if $x\;\R\; x'$ then $x$ and $x'$ are parallel,

\item if $x \;\R\; x'$, $y \;\R\; y'$ and if $x$ and $x'$ are $k$-composable for some $0\leq k <n$ then

\[

x \comp_k x' \;\R\; y\comp_k y'

\]

(which makes sense since $y$ and $y'$ are $k$-composable by the previous axiom).

\end{enumerate}

\end{definition}

\begin{remark}

Beware that in the previous definition, the relation $\R$ is \emph{only} on the set of cells of dimension $n$.

\end{remark}

\begin{example}

Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$. The binary relation $\R$ on $X_n$ defined as

\[

x\R y \text{ if } F(x)=F(y)

\]

is a congruence.

\end{example}

\begin{paragr}

Let $X$ be an $n$-magma with $n \geq1$ and $\R$ a congruence on $X$. By the first axiom of Definition \ref{def:congruence}, $\R$ is an equivalence relation and we can consider the quotient set $X_n/\R$. We write $[x]$ for the equivalence class of an $n$-cell $x$ of $X$. From the second axiom of Definition \ref{def:congruence}, we can define unambiguously

\[

\src([x]):=\src(x)\text{ and }\trgt([x]):=\trgt(x),

\]

for $x \in X_n$ and from the third axiom, we can define unambiguously

\[

[x]\comp_k[y] :=[x \comp_k y]

\]

for $x$ and $y$$k$-composable $n$-cells of $X$. Altoghether, this defines an $n$-magma, which we denote by $X/{\R}$, whose set $k$-cells is $X_k$ for $0\leq k < n$, and $X_n/{\R}$ for $k=n$. The composition, source, target and units of cells of dimension strictly lower than $n$ being those of $X$ and the composition, source and target of $n$-cells being given by the above formulas.

\end{paragr}

\begin{definition}

Let $\R$ be a congruence on an $n$-magma $X$ with $n \geq1$. We say that $\R$ is \emph{categorical} if it satisfies all four following axioms:

\begin{enumerate}

\item for every $k<n$ and every $n$-cell $x$ of $X$, we have

\[

\1^{n}_{\trgt_k(x)}\comp_kx\;\R\; x \text{ and } x \:\R\; x \comp_k\1^n_{\src_k(x)},

\]

\item for every $k<n$ and for all $k$-composable $n$-cells $x$ and $y$ of $X$, we have

\[

1_{x\comp_k y}\;\R\;1_{x}\comp_k 1_{y},

\]

\item for every $k<n$, for all $n$-cells $x, y$ and $z$ of $X$ such that $x$ and $y$ are $k$-composable, and $y$ and $z$ are $k$-composable, we have

\begin{example} Let $C$ be an $n$-category with $n>1$, which we consider as an $n$-magma. The equality on the set of $n$-cells of $C$, is, by definition, categorical.

\end{example}

\begin{example}

Another important example of categorical congruence is the following. Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$ and suppose that $Y$ is an $n$-category. Then the binary relation $\R$ on $X_n$ defined as

\[

x \R y \text{ if } F(x)=F(y)

\]

is obviously a \emph{categorical} congruence.

\end{example}

In the following lemma, we use the notation $\tau_{\leq n}^s(X)$ for an $n$-mag