### I have got to finishls

parent 2d470808
 ... ... @@ -920,16 +920,23 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati \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 $\oo\Cat$ \cite[Proposition 9]{lafont2010folk}, it suffices to show that $\tau^{i}_{\leq n}$ sends folk trivial fibrations of $\oo\Cat$ to weak equivalences of $n\Cat$ (in vertue of Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model}). % 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 $\eta_C(x)\simeq_{\oo}\eta_C(y)$. For convenience let By definition of folk weak equivalences on $n\Cat$, we have to show that $\iota_n\tau^{i}_{\leq n}(f) : \iota_n\tau^{i}_{\leq n}(C) \to \iota_n\tau^{i}_{\leq n}(D)$ is again a weak equivalence for the folk model structure on $\oo\Cat$. Consider the following commutative square is a folk weak equivalence on $\oo\Cat$. Consider the following commutative square $\begin{tikzcd} C \ar[d,"\eta_C"] \ar[r,"f"] & D \ar[d,"\eta_D"] \\ ... ... @@ -938,8 +945,13 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati$ where $\eta$ is the unit of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. \begin{enumerate}[label=(\roman*)] \item Let $y$ be a $0$\nbd-cell of $\iota_n\tau^{i}_{\leq n}(D)$. The map $\eta_D$ being surjective on $0$\nbd-cells (even if $n=0$), there exists $y'$ such that $\eta_D(y')=y$. Since $f$ is a folk weak equivalence, there exists $x' \in C_0$ such that $x'\simeq_{\oo} y'$ and then $x:=\eta_C(x') \simeq_{\oo} y'$. \item Let $x$ and $y$ be parallel \item Let $y$ be a $0$\nbd-cell of $\iota_n\tau^{i}_{\leq n}(D)$. The map $\eta_D$ being surjective on $0$\nbd-cells (even if $n=0$), there exists $y'$ such that $\eta_D(y')=y$. Since $f$ is a folk trivial fibration, there exists $x' \in C_0$ such that $f(x')=y'$ and then if we set $x:=\eta_C(x')$, we have $\iota_n\tau^{i}_{\leq n}(f)(x)=y$. \item Let $x$ and $y$ be parallel $k$\nbd-cells with \$0
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