@@ -257,7 +257,8 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
&\simeq\Hom_{\Set}(\Sigma_n,\vert G \vert)\\
&\simeq\Hom_{\Ab}(\mathbb{Z}\Sigma_n,G),
\end{align*}
and it is easily checked that this isomorphism is induced by the map
and it is easily checked that this isomorphism is induced by
precomposition with the map
$\mathbb{Z}\Sigma_n \to\lambda_n(C)$ from the previous paragraph. The
result follows then from the Yoneda Lemma.
\end{proof}
...
...
@@ -338,7 +339,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\]
Recall also that if there is a chain homotopy from $f$ to $g$, then the
localization functor $\gamma^{\Ch} : \Ch\to\ho(\Ch)$ identifies $f$ and
$g$, meaning that \[\gamma^{\Ch}(f)=\gamma^{\Ch}(g).\]
$g$, which means that \[\gamma^{\Ch}(f)=\gamma^{\Ch}(g).\]
\end{paragr}
%For the definition of \emph{homotopy of chain complexes} see for example \cite[Definition 1.4.4]{weibel1995introduction} (where it is called \emph{chain homotopy}).
\begin{lemma}\label{lemma:abeloplax}
...
...
@@ -1114,12 +1115,12 @@ We now turn to truncations of chain complexes.
\begin{proposition}
There exists a model structure on $\Ch^{\leq n}$ such that:
\begin{itemize}[label=-]
\item weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$,
\item fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\itemthe weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$,
\itemthe fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\end{itemize}
\end{proposition}
\begin{proof}
This is a typical example of a transfer of a cofibrantly generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square
This is a typical example of a transfer of a cofibrantly generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j \colon A \to B$ in $J$ and every cocartesian square
\[
\begin{tikzcd}
\tau^{i}_{\leq n}(A)\ar[r]\ar[d,"\tau^{i}_{\leq n}(j)"']& X \ar[d,"g"]\\
@@ -195,7 +195,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{theorem}
\begin{proof}
Recall from \ref{paragr:nerve} that $c_n : \Psh{\Delta}\to n\Cat$ denotes
the left adjoint to the nerve functor $N_n$. In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta}\to\Psh{\Delta}$, as well as a zigzag of morphisms of functors
the left adjoint of the nerve functor $N_n$. In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta}\to\Psh{\Delta}$, as well as a zigzag of morphisms of functors
