@@ -641,7 +641,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\[
H^{\pol}_k(C)=\begin{cases}\mathbb{Z}\text{ if } k=0,2\\0\text{ in other cases.}\end{cases}
\]
On the other hand, it is proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism).
On the other hand, it is proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism).
\end{paragr}
Another consequence of the above counter-example is the following result, which we claimed in \ref{paragr:polhmlgythomeq}. Recall that given a morphism $u : C \to D$ of $\oo\Cat$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$.
@@ -1130,7 +1130,7 @@ We now turn to truncations of chain complexes.
\]
The isomorphism being obviously induced by the unit map $K \to\iota_n\tau^{i}_{\leq n}(K)$.
\end{proof}
As a consequence of this lemma, we have the analoguous of Proposition \ref{prop:truncationhomotopical}.
As a consequence of this lemma, we have the analogous of Proposition \ref{prop:truncationhomotopical}.
\begin{proposition}
The functor $\tau^{i}_{\leq n} : \Ch\to\Ch^{\leq n}$ sends weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.
\end{proposition}
...
...
@@ -1216,7 +1216,7 @@ is commutative.
\]
Then, it follows from an argument similar to the proof of Lemma \ref{lemma:nfunctortomonoid} (see also the proof of Lemma \ref{lemma:adjlambdasusp}) that this last set is naturally isomorphic to the set of functions $f_n : C_n \to G$ such that:
\begin{itemize}[label=-]
\item for every $0\leq k <n $ and every pair $(x,y)$ of $k$-composables$n$\nbd{}cells of $C$, we have
\item for every $0\leq k <n $ and every pair $(x,y)$ of $k$-composable $n$\nbd{}cells of $C$, we have
\[
f(x \comp_k y)= f(x)+ f(y),
\]
...
...
@@ -1285,7 +1285,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi
for every $0\leq k \leq n$.
\end{corollary}
\begin{proof}
From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morhism $\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$ of $\ho(\Ch^{\leq n})$ can be identified with the canonical morphism
From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morphism $\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$ of $\ho(\Ch^{\leq n})$ can be identified with the canonical morphism
for every $p \geq2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every $k \geq3$, the (nerve of the) $\oo$\nbd{}category $B^k\mathbb{N}$ is a $K(\mathbb{Z},k)$-space. In particular, we have
for every $p \geq2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every $k \geq3$, the (nerve of the) $\oo$\nbd{}category $B^k\mathbb{N}$ is a $K(\mathbb{Z},k)$. In particular, we have
@@ -632,7 +632,7 @@ Furthermore, this function satisfies the condition
is injective.
\end{lemma}
\begin{proof}
A thorough reading of thet echniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ as the amalgamated sum \eqref{squarefreecext} is sufficient enough to prove the existence, for each $x \in\Sigma$, of a function
A thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ as the amalgamated sum \eqref{squarefreecext} is sufficient enough to prove the existence, for each $x \in\Sigma$, of a function
\[
w_{x} : (\E^*)_{n+1}\to\mathbb{N}
\]
...
...
@@ -1855,9 +1855,9 @@ We end this section with yet another characterisation of $n$-basis of $\oo$\nbd{
Let $n>0$, $C$ be an $n$\nbd{}category with an $n$\nbd{}basis $\Sigma$ and
$\alpha\in\Sigma$. It is immediate to check from the definition of
elementary moves that for two equivalent well formed words $u \sim u'$ of
$\T[\Sigma]$, the number of occurences of $\cc_{\alpha}$ in $u$ and $u'$ are
$\T[\Sigma]$, the number of occurrences of $\cc_{\alpha}$ in $u$ and $u'$ are
the same. In particular, for any $a \in C_n$, we can define the integer
$w_{\alpha}(a)$ to be the number of occurences of $\cc_{\alpha}$ in any well
$w_{\alpha}(a)$ to be the number of occurrences of $\cc_{\alpha}$ in any well
formed word $u$ such that $\rho_{\sigma}(u)=a$. An immediate induction using
the properties of $\rho_{\Sigma}$ shows that this function $w_{\alpha} : C_n \to\mathbb{N}$ is the same
as the one whose existence was established in Proposition
