### 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 ... ...
 ... @@ -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
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