edited a typo and added an argument in a proof

hmlgy.tex

@@ -1107,7 +1107,7 @@ We now turn to truncations of chain complexes.
   Again, as with $n$\nbd{}categories, we can use the adjunction
   \[
   \begin{tikzcd}
-    \tau^{i}_{\leq n} : \Ch^{\leq n}\ar[r,shift left] & \ar[l,shift left]\Ch : \iota_n 
+    \tau^{i}_{\leq n} : \Ch\ar[r,shift left] & \ar[l,shift left]\Ch^{\leq n} : \iota_n 
   \end{tikzcd}
   \]
   to create a model structure on $\Ch^{\leq n}$.
@@ -1120,7 +1120,10 @@ We now turn to truncations of chain complexes.
     \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 $j \colon A \to B$ 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}. Since the weak equivalences of the projective model
+  structure on $\Ch$ are closed under filtered colimits \cite[Theorem 2.6.15]{weibel1995introduction}, 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"] \\