@@ -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 $u \in\W[\Sigma^C]$:
