Commit d9419d8c authored by Leonard Guetta's avatar Leonard Guetta
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security commit

parent 55b7a9eb
......@@ -575,13 +575,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]
\end{tikzcd}
\]
A throrough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following descritption of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$
%% \begin{tikzcd}
%% c_{\oo} : \Psh{\Delta} \ar[r,shift left] &\oo\Cat : N_{\oo}\ar[l,shift left]
%% \end{tikzcd}
with the abelianization functor, we obtain $2$-morphism
A throrough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following descritption of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism
\[
\lambda c_{\oo} N_{\oo} \Rightarrow \lambda.
\]
......@@ -1141,7 +1135,7 @@ As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n}
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[d,"\overline{\tau_{\leq n}^{i}}"] \ar[r,"\LL \lambda"] & \ho(\Ch) \ar[d,"\overline{\tau^{i}_{\leq n}}"] \\
\ho(n\Cat^{\Th}) \ar[r,"\LL \lambda_{\leq n}"] & \ho(\Ch^{\leq n})
\ho(n\Cat^{\folk}) \ar[r,"\LL \lambda_{\leq n}"] & \ho(\Ch^{\leq n})
\end{tikzcd}
\]
is commutative (up to a canonical isomorphism).
......@@ -1225,7 +1219,7 @@ Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and th
\]
which is natural in $C$.
\end{paragr}
\begin{lemma}
\begin{lemma}\label{lemma:truncationcounit}
For every $\oo$\nbd-category $C$, the canonical morphism of $\Cat$
\[
c_1N_{\oo}(C) \to \tau^{i}_{\leq 1}(C)
......@@ -1255,7 +1249,7 @@ Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and th
\]
in $C$, we have $F_1(g)\comp_0 F_1(f)=F_1(h)$.
\end{itemize}
In particular, it follows that $F_1$ is compatible with composition of $1$\nbd-cells in an obvious sense and that for every $2$\nbd-cell $\alpha : f \Rightarrow g$ of $C$, we have $F_1(f)=F_2(g)$. This means exactly that we have a natural isomorphism
In particular, it follows that $F_1$ is compatible with composition of $1$\nbd-cells in an obvious sense and that for every $2$\nbd-cell $\alpha : f \Rightarrow g$ of $C$, we have $F_1(f)=F_1(g)$. This means exactly that we have a natural isomorphism
\[
\Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\oo\Cat}(\tau_{\leq 1}^{i}(C),D).
\]
......@@ -1282,7 +1276,19 @@ We can now prove the important following proposition.
for $k \in \{0,1\}$.
\end{proposition}
\begin{proof}
\todo{À écrire}
Let $C$ be an $\oo$\nbd-category. Recall from \ref{paragr:univmor} that the canonical morphism $\alpha^{\sing} : \sH^{\sing}(C) \to \lambda(C)$ is nothing but the image by the localization functor $\Ch \to \ho(\Ch)$ of the morphism
\[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\]
induced by the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$. From \ref{prop:polhmlgytruncation} we have that
\[
\tau^{i}_1\lambda c_{\oo}N_{\oo}(C) \simeq \lambda_{\leq 1} \tau_{\leq 1}^{i} c_{\oo} N_{\oo}(C)=\lambda_{\leq 1} c_1 N_{\oo}(C),
\]
and from Lemma \ref{lemma:truncationcounit} we obtain
\[
\tau_{\leq 1 }^{i} \lambda c_{\oo} N_{\oo}(C) \simeq \lambda_{\leq 1} \tau^{i}_{\leq 1}(C) \simeq \tau^{i}_{\leq 1}\lambda(C).
\]
This means exactly that the image by $\overline{\tau^{i}_{\leq 1}}$ of $\alpha^{\sing}$ is an isomorphism, which is what we wanted to prove.
\end{proof}
Finally, we obtain the result we were aiming for.
\begin{proposition}
......
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