Commit 81e86938 authored by Leonard Guetta's avatar Leonard Guetta
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tpoto

parent 18fede95
......@@ -386,7 +386,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd-funct
Let $A$ a $1$\nbd-category, $X$ a free $\oo$\nbd-category and $f : X \to A$ be an $\oo$\nbd-functor. The functor
\begin{align*}
A &\to \oo\Cat \\
a_0 &\mapsto A/a_0
a_0 &\mapsto X/a_0
\end{align*}
is a cofibrant object for the projective model structure on $\oo\Cat(A)$
induced by the folk model structure on $\oo\Cat$ (\ref{paragr:projmod}).
......
......@@ -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.
\end{paragr}
\begin{proposition}\label{prop:truncationhomotopical}
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
\[
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......@@ -179,13 +179,13 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
The sequence of adjunctions
\[
\kappa_n \dashv \tau^s_{\leq n} \dashv \iota_n \dashv \tau^{i}_{\leq n}
\tau^{i}_{\leq n} \dashv \iota_n \dashv \tau^s_{\leq n} \dashv \kappa_n
\]
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 $n \geq 0$, 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.
\end{enumerate}
\end{paragr}
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