Commit 9bd4321c authored by Leonard Guetta's avatar Leonard Guetta
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parent 033c16d3
......@@ -902,7 +902,7 @@ The previous proposition admits the following corollary, which will be of great
\]
In fact, we can use the adjunction $\tau_{\leq n}^{i} \dashv \iota_n$ to transport the folk model structure on $\oo\Cat$ to $n\Cat$.% Say that a morphism $f : C \to D$ of $n\Cat$ is a \emph{folk weak equivalence of $n$\nbd-categories} (resp. \emph{folk trivial fibration of $n$\nbd-categories}) if $\iota_n(f)$ is a weak equivalence (resp.\ fibration) for the folk model structure on $\oo\Cat$.
\end{paragr}
\begin{proposition}
\begin{proposition}\label{prop:fmsncat}
There exists a model structure on $n\Cat$ such that:
\begin{itemize}[label=-]
\item weak equivalences are exactely those morphisms $f : C \to D$ such that $\iota_n(f)$ is a weak equivalence for the folk model structure on $\oo\Cat$,
......@@ -1036,7 +1036,7 @@ As a consequence of this lemma, we have the analoguous of Proposition \ref{prop:
\]
is an isomorphism for every $k \leq n$. Since obviously $H_k(\iota_n\tau^{i}_{\leq n}(f))$ is also an isomorphism for $k > n$, this proves the result.
\end{proof}
We now investigate the relation between truncation and linearization.
We now investigate the relation between truncation and abelianization.
\begin{paragr}
Let $C$ be $n$\nbd-category. A straigtforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that
\[
......@@ -1056,7 +1056,7 @@ We now investigate the relation between truncation and linearization.
\]
is commutative.
\end{paragr}
\begin{lemma}
\begin{lemma}\label{lemma:abelianizationtruncation}
The square
\[
\begin{tikzcd}
......@@ -1123,8 +1123,44 @@ is commutative.
%% \tau^{i}_{\leq n}(
%% \]
\end{proof}
With this lemma at hand we can prove the important following proposition which basically says that if an $\oo$\nbd-category $C$ is free up to dimension $n-1$, then for any $k$ such that $0 \leq k \leq n$ there is no need to find a cofibrant replacement in order to compute $H^{\pol}_k(C)$.
\begin{proposition}
\begin{lemma}
The functor $\lambda_{\leq n} : n\Cat \to \Ch^{\leq n}$ is left Quillen when $n\Cat$ is equipped with the folk model structure and $\Ch^{\leq n}$ with the projective model structure.
\end{lemma}
\begin{proof}
Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$. We know from Proposition \ref{prop:fmsncat} that $\tau^{i}_{\leq n}(I)$ and $\tau^{i}_{\leq n}(J)$ respectively are sets of generating cofibrations and generating trivial cofibrations of the projective model structure on $\Ch^{\leq n}$. What we have to show is that for every $f$ in $I$ (resp. $J$), $\lambda_{\leq n}(\tau^{i}_{\leq n}(f))$ is a cofibration (resp.\ generating cofibration) for the folk model structure on $n\Cat$.
% Notice that we have $\tau^{i}_{\leq n } \circ \iota_n = \mathrm{id}_{n \Cat}$ and hence
From Lemma \ref{lemma:abelianizationtruncation}, we have
\[
\lambda_{\leq n }(\tau^{i}_{\leq n}(f)) \simeq \tau^{i}_{\leq n}(\lambda(f)).
\]
Since $\lambda$ and $\tau^{i}_{\leq n}$ are both left Quillen functors, this proves the result.
\end{proof}
As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n}$ is left derivable and we have the following key result.
\begin{proposition}\label{prop:polhmlgytruncation}
The square
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[d,"\overline{\tau_{\leq n}^{i}}"] \ar[r,"\LL \lambda"] & \ho(\Ch) \ar[d,"\overline{\tau^{i}_{\leq n}}"] \\
\ho(n\Cat^{\Th}) \ar[r,"\LL \lambda_{\leq n}"] & \ho(\Ch^{\leq n})
\end{tikzcd}
\]
is commutative (up to a canonical isomorphism).
\end{proposition}
\begin{proof}
Straightforward consequence of Lemma \ref{lemma:abelianizationtruncation} and the fact that the left derived functor of a composition of left Quillen functors is the composition of the left derived functors (see for example \cite[Theorem 1.3.7]{hovey2007model}.
\end{proof}
\begin{remark}
Beware that the square
\[
\begin{tikzcd}
\ho(n\Cat^{\folk}) \ar[d,"\overline{\iota_n}"] \ar[r,"\LL \lambda_{\leq n}"] & \ho(\Ch^{\leq n}) \ar[d,"\overline{\iota_n}"] \\
\ho(\oo\Cat^{\folk}) \ar[r,"\LL \lambda"] & \ho(\Ch)
\end{tikzcd}
\]
is \emph{not} commutative. If it were, then for every $n$\nbd-category $C$ and every $k >n$, we would have $H_k^{pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$ in the following chapter.
\end{remark}
A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the following corollary.
\begin{corollary}
Let $n \geq 0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $ 0 \leq k \leq n-1$, then the canonical map of $\ho(\Ch)$
\[
\alpha^{\pol}_C : \sH^{\pol}(C) \to \lambda(C)
......@@ -1134,28 +1170,36 @@ With this lemma at hand we can prove the important following proposition which b
H_k^{\pol}(C) \simeq H_k(\lambda(C))
\]
for every $0 \leq k \leq n$.
\end{proposition}
\end{corollary}
\begin{proof}
\todo{À écrire}
\end{proof}
\begin{paragr}
Since every $\oo$\nbd-category $C$ admits $C_0$ as a $0$\nbd-base, it follows from the previous proposition that $H^{\pol}_0(C)$ and $H^{\pol}_1(C)$ respectively are (canonically isomorphic to) $H_0(\lambda(C))$ and $H_1(\lambda(C))$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
\end{paragr}
Somewhat related is the following proposition.
\begin{proposition}
Let $C$ be an $\oo$\nbd-category and $n \geq 0$. The canonical map
Since every $\oo$\nbd-category trivially admits its set of $0$\nbd-cells as a $0$\nbd-base, it follows from the previous proposition that for every $\oo$\nbd-category $C$ we have
\[
\sH^{\pol}(C) \to \sH^{\pol}(\iota_n\tau^{i}_{\leq n}(C))
\sH^{\pol}_0(C)\simeq H_0(\lambda(C))
\]
induces isomorphisms
and
\[
H^{\pol}_k(C) \simeq H^{\pol}_k(\iota_n\tau^{i}_{\leq n}(C))
\sH^{\pol}_1(C) \simeq H_1(\lambda(C)).
\]
for every $0 \leq k \leq n$.
\end{proposition}
\begin{proof}
Let $f : P \to C$ be a cofibrant replacement for $C$. \todo{À finir}.
\end{proof}
Intuitively speaking, this means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
\end{paragr}
%% Somewhat related is the following proposition.
%% \begin{proposition}
%% Let $C$ be an $\oo$\nbd-category and $n \geq 0$. The canonical map
%% \[
%% \sH^{\pol}(C) \to \sH^{\pol}(\iota_n\tau^{i}_{\leq n}(C))
%% \]
%% induces isomorphisms
%% \[
%% H^{\pol}_k(C) \simeq H^{\pol}_k(\iota_n\tau^{i}_{\leq n}(C))
%% \]
%% for every $0 \leq k \leq n$.
%% \end{proposition}
%% \begin{proof}
%% Let $f : P \to C$ be a cofibrant replacement for $C$. \todo{À finir}.
%% \end{proof}
We now turn to the relation between truncation and singular homology of $\oo$\nbd-categories. Recall that for any $n \geq 0$, the nerve functor $N_n : n\Cat \to \Psh{\Delta}$ is defined as the following composition
\[
N_n : n\Cat \overset{\iota_n}{\longrightarrow} \oo\Cat \overset{N_{\oo}}{\longrightarrow} \Psh{\Delta},
......@@ -1172,7 +1216,7 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
is commutative (up to an isomorphism).
\end{lemma}
\begin{proof}
\todo{À écrire (facile)}
Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and the fact that the composition of left adjoints if the left adjoint of the composition.
\end{proof}
\begin{paragr}
In particular, it follows from the previous lemma that the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ induces, for every $\oo$\nbd-category $C$ and every $n \geq 0$, a canonical morphism of $n\Cat$
......@@ -1189,7 +1233,16 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
is an isomorphism.
\end{lemma}
\begin{proof}
\todo{À écrire.}
Let $C$ be an $\oo$\nbd-category and $D$ a (small) category. By adjunction, we have
\[
\Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)).
\]
Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq 2} \to \Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$. Recall that the nerve of a (small) category is $2$-coskelettal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
\[
\Hom_{\oo\Cat}(N_{\oo}(C),N_1(D)) \simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
\]
Using
% It might be useful now to recall
\end{proof}
We can now prove the important following proposition.
\begin{proposition}
......
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