Commit 7f52396e authored by Leonard Guetta's avatar Leonard Guetta
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Dirty merging with the impression branch

parent b0471cb5
...@@ -92,8 +92,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends the weak e ...@@ -92,8 +92,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends the weak e
definition of homology. This assertion will be justified in Remark \ref{remark:polhmlgyisnotinvariant}. definition of homology. This assertion will be justified in Remark \ref{remark:polhmlgyisnotinvariant}.
\end{remark} \end{remark}
\begin{remark} \begin{remark}
We could also have defined the singular homology of $\oo$\nbd{}categories We could also have defined the singular homology of $\oo$\nbd{}categories using $K : \Psh{\Delta}\to \Ch$ instead of $\kappa : \Psh{\Delta} \to \Ch$ since these two functors are quasi-isomorphic (see \cite[Theorem 2.4]{goerss2009simplicial} for example). An advantage of the latter one is that it is left Quillen.
using $K : \Psh{\Delta}\to \Ch$ instead of $\kappa : \Psh{\Delta} \to \Ch$ since these two functors are quasi-isomorphic (see \cite[Theorem 2.4]{goerss2009simplicial} for example). An advantage of the latter one is that it is left Quillen.
\end{remark} \end{remark}
\begin{paragr} \begin{paragr}
We will also denote by $\sH^{\sing}$ the morphism of op-prederivators defined as the following composition We will also denote by $\sH^{\sing}$ the morphism of op-prederivators defined as the following composition
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...@@ -799,7 +799,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen ...@@ -799,7 +799,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
$c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k$,&for every $k+1 \leq i \leq n$.\\ $c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k$,&for every $k+1 \leq i \leq n$.\\
\end{tabular} \end{tabular}
\end{itemize} \end{itemize}
We leave it to the reader to check that the formulas are well defined and that the axioms of $\oo$\nbd{}category are satisfied. The canonical forgetful $\oo$\nbd{}functor $\pi : A/a_0 \to A$ is simply expressed as: We leave it to the reader to check that the formulas are well defined and that the axioms for $\oo$\nbd{}categories are satisfied. The canonical forgetful $\oo$\nbd{}functor $\pi : A/a_0 \to A$ is simply expressed as:
\begin{align*} \begin{align*}
A/a_0 &\to A \\ A/a_0 &\to A \\
(x,a) &\mapsto x_n. (x,a) &\mapsto x_n.
...@@ -831,7 +831,15 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen ...@@ -831,7 +831,15 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
& (x_n,b_{n+1}) & (x_n,b_{n+1})
\end{pmatrix} \end{pmatrix}
\] \]
where the $x_i$ and $x'_i$ are $i$-cells of $A$, and the $b_i$ and $b'_i$ are $i$-cells of $B$, such that where the $x_i$ and $x'_i$ are $i$-cells of $A$ such that
\begin{tabular}{ll}
$x_i : x_{i-1} \longrightarrow x'_{i-1}$, &for every $1 \leq i \leq n$,\\[0.75em]
$x_i': x_{i-1} \longrightarrow x'_{i-1}$, &for every $1 \leq i \leq n-1$,\\[0.75em]
\end{tabular}
and the $b_i$ and $b'_i$ are $i$-cells of $B$ such that
\[ \[
\begin{pmatrix} \begin{pmatrix}
\begin{matrix} \begin{matrix}
...@@ -890,7 +898,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen ...@@ -890,7 +898,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
Let us use the notations: Let us use the notations:
\begin{itemize}[label=-] \begin{itemize}[label=-]
\item $a_i := \src_i(f)=\src_i(f')$ for $0 \leq i <n$, \item $a_i := \src_i(f)=\src_i(f')$ for $0 \leq i <n$,
\item $a_i := \trgt_i(f)=\trgt_i(f')$ for $0 \leq i <n$, \item $a_i' := \trgt_i(f)=\trgt_i(f')$ for $0 \leq i <n$,
\item $a_n:=f$ and $a_n'=f'$. \item $a_n:=f$ and $a_n'=f'$.
\end{itemize} \end{itemize}
It is straightforward to check that It is straightforward to check that
...@@ -933,7 +941,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen ...@@ -933,7 +941,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
& (\alpha,\Lambda) & (\alpha,\Lambda)
\end{pmatrix} \end{pmatrix}
\] \]
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above ${(n+1)}$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0 \to A$. whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above ${(n+1)}$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $u(\alpha) \sim_{\oo} \beta$ by applying the canonical $\oo$\nbd{}functor $B/c_0 \to B$.
\end{enumerate} \end{enumerate}
\end{proof} \end{proof}
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