@@ -586,7 +586,7 @@ For later reference, we put here the following trivial but important lemma, whos

An $\omega$-category is cofibrant for the folk model structure if and only if it is free.

\end{proposition}

\begin{proof}

The fact that every free $\omega$-category is cofibrant follows immediately from the fact that the $i_n : \sS_{n-1}\to\sD_n$ are cofibrations and that every $\omega$-category $C$ is the colimit of the canonical diagram (Lemma \ref{lemma:filtration})

The fact that every free $\omega$-category is cofibrant follows immediately from the fact that the $i_n : \sS_{n-1}\to\sD_n$ are cofibrations and that every $\oo$\nbd{}category $C$ is the colimit of the canonical diagram (Lemma \ref{lemma:filtration})

Intuitively speaking, $\rho_{\Sigma}$ is to be understood as an ``evaluation

map'': for every$w$ well formed expression on units and formal cells of

map'': given$w$a well formed expression on units and formal cells of

$\Sigma$, $\rho_{\Sigma}(w)$ is the evaluation of $w$ as an $(n+1)$\nbd{}cell

of $C$.

\end{paragr}

...

...

@@ -1973,7 +1973,8 @@ with

\item[-]$\wt{F}(\,(\,)=($,

\item[-]$\wt{F}(\,)\,)=)$.

\end{itemize}

Notice that for every word $w \in\W[\Sigma^C]$, we have $|\wt{F}(w)|=|w|$ and $\mathcal{L}(\wt{F}(w))=\mathcal{L}(w)$.

Notice that for every word $w \in\W[\Sigma^C]$, we have

\[\vert\wt{F}(w)\vert=\vert w \vert\text{ and }\mathcal{L}(\wt{F}(w))=\mathcal{L}(w).\]

\end{paragr}

\begin{lemma}\label{lemmamapinducedonwords}

Let $F : C \to D$ be an $\omega$-functor, $\Sigma^C \subseteq C_{n}$ and $\Sigma^D \subseteq D_{n}$ such that $F_{n}(\Sigma^C)\subseteq\Sigma^D$. For every $u \in\W[\Sigma^C]$: