Commit 17ea8409 authored by Leonard Guetta's avatar Leonard Guetta
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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 $0<k<n-1$, since
$\eta_C$ and $\eta_D$ are the identities on $p$-cells for eveyr $0<p<$
\end{enumerate}
\todo{À finir}
\end{proof}
......@@ -1358,3 +1370,8 @@ Finally, we obtain the result we were aiming for.
%% \end{lemma}
%%\section{Homology and Homotopy of $\oo$-categories in low dimension}
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