@@ -610,7 +610,7 @@ Furthermore, this function satisfies the condition

(\E^*)_{n+1}\ar[r,"(G^*)_{n+1}"]&(\E'^*)_{n+1}

\end{tikzcd}

\]

is commutative. Notice also that $j$ is compatible with source and targets in the sense that for every $x \in\Sigma$, we have

is commutative. Notice also that $j$ is compatible with source and target in the sense that for every $x \in\Sigma$, we have

\[

\src(j(x))=\sigma(x)\text{ and }\trgt(j(x))=\tau(x).

\]

...

...

@@ -629,9 +629,14 @@ Furthermore, this function satisfies the condition

\]

such that $w_{x}(j(x))=1$ and $w_{x}(j(y))=0$ for any $y \in\Sigma$ with $y\neq x$. In particular, this implies that $j$ is injective.

\end{proof}

\begin{paragr}

In consequence of the previous lemma, we will often identify $\Sigma$ to a subset of $(\E^*)_{n+1}$. When we do so, it will \emph{always} be via tha map $j$. Note that this identification is compatible with source and target in the sense that the source (resp. target) of $x \in\Sigma$, seen as $(n+1)$-cell of $\E^*$, is exactly $\sigma(x)$ (resp. $\tau(x)$).

\end{paragr}

\iffalse

\begin{paragr}

In particular, the previous lemma tells us that for any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the set of indeterminates $\Sigma$ canonically defines a subset of the $(n+1)$-cells of $\E^*$. As of now, we will consider $\Sigma$ as a subset of $(\E^{*})_{n+1}$ and $j$ as the canonical inclusion.

\end{paragr}

\fi

We can now prove the following proposition, which is the key result of this section. It is slightly less trivial than it appears.

\begin{proposition}\label{prop:fromcexttocat}

For any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the subset $\Sigma\subseteq(\E^{*})_{n+1}$ is an $(n+1)$-basis of $\E^*$.

...

...

@@ -809,9 +814,11 @@ Recall that an $n$-category is a particular case of $n$-magma.

\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).

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{definition}\label{def:sizeword}

The \emph{size} of a well-formed word $w$, denoted by $\vert w \vert$, is the number of symbols $\fcomp_k$ for any $0\leq k \leq n$ that appear in $w$.

\end{definition}

\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=-]

...

...

@@ -893,8 +900,8 @@ In the following definition, we consider that a binary relation $\R$ on a set $E

\end{definition}

\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 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

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)

\]

...

...

@@ -902,8 +909,8 @@ In the following definition, we consider that a binary relation $\R$ on a set $E

\end{example}

Similarly to the ``stupid'' truncation of $\oo$-categories (\ref{paragr:defncat}), for an $(n+1)$-magma $X$, we write $\tau_{\leq n}^s(X)$ for the $n$-magma obtained by simply forgetting the cells of dimension $(n+1)$.

The following lemma is trivial but nonetheless important. Its immediate proof is omitted.

\begin{lemma}

The following lemma is trivial but nonetheless important. Its immediate proof is ommited.

\begin{lemma}\label{lemma:quotientmagma}

Let $X$ be an $n$-magma with $n>1$ and $\R$ a congruence on $X$. If $\tau_{\leq n}^s(X)$ is an $(n-1)$-category and $\R$ is categorical, then $X/{\R}$ is an $n$-category.

\end{lemma}

We wish now to see how to prove the existence of a congruence defined with a condition such as ``the smallest congruence that contains a given binary relation on the $(n+1)$-cells''.

...

...

@@ -940,6 +947,110 @@ We wish now to see how to prove the existence of a congruence defined with a con

\begin{proof}

Each four axioms of Definition \ref{def:categoricalcongruence} says that some pairs of parallel $n$-cells must be equivalent under a congruence $\R$ for it to be categorical. The result follows then from Lemma \ref{lemma:congruencegenerated}.

\end{proof}

\iffalse\begin{paragr}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension, $D$ an $(n+1)$-category and

\[

G=(\varphi,F) : \E\to U_n(D)

\]

a morphism of $n$-cellular extensions.

\end{paragr}

\fi

\begin{paragr}

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension and consider the injective map $j : \Sigma\to(\E^*)_{n+1}$ constructed in Paragraph \ref{paragr:freecext}. By induction, we define a map

In particular, for $\E$ an $n$-cellular extension, there exists a smallest categorical congruence on $\E^+$, which we denote here by $\equiv$. Applying Lemma \ref{lemma:quotientmagma} gives us that $\E^+/{\equiv}$ is an $(n+1)$-category. The construction \[\E\mapsto\E^+/{\equiv}\] clearly defines a functor $\Cat_n^+\to(n+1)\Cat$.

\end{paragr}

We can now prove the expected result.

\begin{proposition}

Let $\E$ be an $n$-cellular extension and let $\equiv$ be the smallest categorical congruence on $\E^+$. The $(n+1)$-category $\E^+/{\equiv}$ is naturally isomorphic to $\E^*$.

\end{proposition}

\begin{proof}

The strategy of the proof is to show that the functor $\E\mapsto\E^+/{\equiv}$ is left adjoint to $U_n : (n+1)\Cat\to\Cat_n^+$. The result will follow then from the uniqueness (up to natural isomorphism) of left adjoints.

Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension, $D$ an $(n+1)$-category and let

\[

G=(\varphi,F) : \E\to U_n(D)

\]

be a morphism of $n$-cellular extensions. We recursively define a map

\[

\overline{\varphi} : \T[\E]\to D_{n+1}

\]

as:

\begin{itemize}[label=-]

\item$\overline{\varphi}(\cc_{\alpha})=\varphi(\alpha)$ for $\alpha\in\Sigma$,

\item$\overline{\varphi}(\ii_x)=1_{F(x)}$ for $x \in C_n$,

A straightforward induction shows that $\overline{\varphi}$ is compatible with source, target and units. Hence, we can define a morphism of $(n+1)$-magmas

\[

\overline{G} : \E^+\to D

\]

as

\[

\overline{G}_{n+1}=\overline{\varphi}

\]

and

\[

\overline{G}_k=F_k \text{ for }0\leq k \leq n.

\]

Since $D$ is an $(n+1)$-category, the binary relation $\R$ on $\T[\E]$ defined as

\[

x\;\R\; y \text{ if }\overline{\varphi}(x)=\overline{\varphi}(y)

\]

is a \emph{categorical} congruence on $\E^+$ (Example \ref{example:categoricalcongruence}). If we denote by $\equiv$ the smallest categorical congruence on $\E^+$ then, by definition, $\equiv$ is included in $\R$. In particular, the map $\overline{\varphi}$ induces a map

\[

\widehat{\varphi} : \T[\E]/{\equiv}\to D_{n+1}.

\]

Since $\equiv$ is a congruence, this map is compatible with source and target, and it is straightforward to check that it is also compatible with units. Hence, we have an $(n+1)$-functor

which is clearly natural in $\E$ and $D$. What we want to prove is that this map is a bijection.

Let us start with the surjectivity. Let $H : \E^+/{\equiv}\to\D$ be a $(n+1)$-functor. We define a map $\varphi : \Sigma\to D_{n+1}$ as

\begin{align*}

\varphi : \Sigma&\to D_{n+1}\\

\alpha&\mapsto H_{n+1}([\cc_{\alpha}]),

\end{align*}

where $[w]$ is the equivalence class under $\equiv$ of an element $w \in\T[\E]$. All we need to show is that

\[

\widehat{\varphi}=H_{n+1}.

\]

Let $z$ be an element of $\T[\E]/{\equiv}$ and let us choose $w \in\T[\E]$ such that $z=[w]$. We proceed to show that $\widehat{\varphi}(z)=H_{n+1}(z)$ by induction on the size of $w$ (Definition \ref{def:sizeword}).

If $|w|=0$, then either $w=\cc_{\alpha}$ for some $\alpha\in\Sigma$ or $w =\ii_x$ for some $x \in C_n$. In the first case, we have

where for the last equality, we used the fact that $[\ii_x]$ is the unit on $x$ in $\E^+/{\equiv}$.

Now if $|w| = n+1$ with $n \geq0$, then $w=(w'\fcomp_k w'')$ for some $w', w'' \in\T[\E]$ that are $k$-composable with $k\leq n$. Hence, using the induction hypothesis, we have

We now turn to the injectivity. Let $G=(\varphi,F)$ and $G'=(\varphi',F')$ be two morphism of $n$-cellular extensions $\E\to U_n(D)$ such that $\widehat{G}=\widehat{G'}$. Since $F=\tau_{\leq n}^s(\widehat{G})=\tau_{\leq n}^s(\widehat{G'})=F'$, all we have to show is that

\[

\varphi=\varphi'.

\]

But, by definition, for every $\alpha\in\Sigma$ we have

In particular, the previous proposition tells us that $n$-cells in a free $\oo$-category can be represented as a (well formed) words made up of the generating $n$-cells and units on lower dimensionan cells.