@@ -908,12 +908,22 @@ The previous proposition admits the following corollary, which will be of great

This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generating cofibrations is not made explicit in the statement of the theorem, it is contained in proof.)

\end{proof}

\begin{paragr}

We refer to the model structure of the above proposition as the \emph{folk model structure on $n\Cat$}. By definition, the functor $\iota_n : n\Cat\to\oo\Cat$ preserves weak equivalences and fibrations when $\oo\Cat$ and $n\Cat$ are equipped with the folk model structure. In particular, the adjunction $\tau^{i}_{\leq n }\dashv\iota_n$ is a Quillen adjunction. As it happens, the functor $\tau^{i}_{\leq n}$ also preserves weak equivalences.

We refer to the model structure of the above proposition as the \emph{folk

model structure on $n\Cat$}. By definition, the functor $\iota_n : n\Cat\to

\oo\Cat$ preserves weak equivalences and fibrations when $\oo\Cat$ and $n\Cat$

are equipped with the folk model structure. In particular, the adjunction

$\tau^{i}_{\leq n }\dashv\iota_n$ is a Quillen adjunction. As it happens,

the functor $\tau^{i}_{\leq n}$ also preserves weak equivalences.

The functor $\tau^{i}_{\leq n} : \oo\Cat\to n\Cat$ sends weak equivalences of the folk model structure on $\oo\Cat$ to weak equivalences of the folk model structure on $n\Cat$.

\end{proposition}

\begin{proof}

Since every $\oo$\nbd-category is fibrant for the folk model structure on

$\oo\Cat$ (see \cite[Proposition 9]{lafont2010folk}), it suffices in vertue of

Ken Brown's Lemma \cite{} to show that

$\tau^{i}_{\leq n}$ sends folk trivial fibrations to weak equivalences of the

folk model structure on $n\Cat$.

% Let $C$ be an $\oo$-category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq n}(C)$ be the unit morphism of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. Let us prove that for every pair $(x,y)$ of $k$\nbd-cells of $C$, if $k \leq n$ then $x \simeq_{\oo} y$ (see \ref{paragr:ooequivalence}) if and only if $\eta_C(x)\simeq_{\oo}\eta_C(y)$.

Let $f : C \to D$ be a weak equivalence for the folk model structure on $\oo\Cat$. By definition of weak equivalences for the the folk model structure on $n\Cat$, we have to show that

is maximal in that $\kappa_n$ doesn't have a left adjoint and $\tau^{i}_{\leq n}$ doesn't have right adjoint.

is maximal in that $\kappa_n$ doesn't have a right adjoint and $\tau^{i}_{\leq n}$ doesn't have left adjoint.

The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}.

The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units on lower dimensional cells.

The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units.

\end{paragr}

\begin{paragr}

For $n \geq0$, we define the $n$-skeleton functor $\sk_n : \oo\Cat\to\oo\Cat$ as

...

...

@@ -2317,3 +2317,8 @@ Putting all the pieces together, we finally have the awaited proof.

Hence, we can apply the second part of Proposition \ref{prop:conduchenbasis} and $C$ is free.