Commit 32d2ff06 by Leonard Guetta

### edited a few typos and corrected the proof of 5.2.3

parent beac890d
 ... ... @@ -1205,7 +1205,7 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}. N_{\oo}(P'') \ar[r] & N_{\oo}(P) \end{tikzcd} \] is also cocartesian. This proves that square \ref{square:bouquet} is is also cocartesian. This proves that square \eqref{square:bouquet} is Thomason homotopy cocartesian \todo{détailler?} and in particular that $P$ has the homotopy type of a bouqet of two $2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are free and \good{} and since $\langle \beta \rangle : \sD_1 \to P'$ and ... ...
 ... ... @@ -166,24 +166,27 @@ From these two lemmas, follows the important proposition below. For every $n \geq -1$, the $\oo$\nbd{}category $\sS_n$ is \good{}. \end{proposition} \begin{proof} We proceed by induction on $n$. When $n=-1$, it is trivial to check that the empty $\oo$\nbd{}category is \good{}. Now, since $i_n : \sS_n \to \sD_{n+1}$ is a folk cofibration and $\sS_{n}$ and $\sD_{n}$ are folk cofibrant, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that the cocartesian square \label{square} %% We proceed by induction on $n$. When $n=-1$, it is trivial to check that the %% empty $\oo$\nbd{}category is \good{}. Recall that the cofibrations of simplicial sets are exactly the monomorphisms, and in particular all simplicial sets are cofibrant. Since $i_n : \sS_{n-1} \to \sD_n$ is a monomorphism for every $n \geq 0$ and since $N_{\oo}$ preserves monomorphisms (as a right adjoint), it follows from Lemma \ref{lemma:squarenerve} and Lemma \ref{lemma:hmtpycocartesianreedy} that the square $\begin{tikzcd} \sS_{n} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_{n+1} \ar[d,"j_n^+"] \\ \sD_{n+1} \ar[r,"j_n^-"] & \sS_{n+1} \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] N_{\oo}(\sS_{n-1}) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n}) \ar[d,"N_{\oo}(j_n^+)"] \\ N_{\oo}(\sD_{n}) \ar[r,"N_{\oo}(j_n^-)"] & N_{\oo}(\sS_{n}) \end{tikzcd} is homotopy cocartesian with respect to the canonical weak equivalences on \oo\Cat. Besides, since N_{\oo} : \oo\Cat \to \Psh{\Delta} induces an$ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an equivalence of op-prederivators $\Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$ (Theorem \ref{thm:gagna}), it follows from Lemma \ref{lemma:squarenerve} that the square \eqref{square} is also homotopy cocartesian with respect to Thomason equivalences. Then, the desired result follows from the induction hypothesis and Corollary \ref{cor:usefulcriterion}. (Theorem \ref{thm:gagna}), it follows that the square of $\oo\Cat$ $\begin{tikzcd} \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_{n} \ar[d,"j_n^+"] \\ \sD_{n} \ar[r,"j_n^-"] & \sS_{n} \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}$ is Thomason homotopy cocartesian for every $n\geq 0$. Finally, since $i_n : \sS_{n-1} \to \sD_{n}$ is a folk cofibration and $\sS_{n-1}$ and $\sD_{n}$ are folk cofibrant for every $n\geq0$, we deduce the result from Corollary \ref{cor:usefulcriterion} and an immediate induction. The base case being simply that $\sS_{-1}=\emptyset$ is obviously \good{}. \end{proof} \begin{paragr} The previous proposition implies what we claimed in Paragraph \ref{paragr:prelimcriteriongoodcat}, which is that the morphism of op-prederivators ... ... @@ -207,7 +210,7 @@ From these two lemmas, follows the important proposition below. $i_2$ is a cofibration for the canonical model structure, the square is also homotopy cocartesian with respect to folk weak equivalences. If $\J$ was homotopy cocontinuous, then this square would also be homotopy cocartesian with respect to Thomason equivalences. Since we know that $\sS_A$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}. with respect to Thomason equivalences. Since we know that $\sS_1$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}. From Proposition \ref{prop:spheresaregood}, we also deduce the proposition below which gives a criterion to detect \good{} $\oo$\nbd{}category when we ... ... @@ -230,7 +233,7 @@ From these two lemmas, follows the important proposition below. \end{proposition} \begin{proof} Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories $\sS^{k-1}$ and $\sS^{k}$ are folk cofibrant and \good{}, it follows $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant and \good{}, it follows from Corollary \ref{cor:usefulcriterion} that all $\sk_k(C)$ are \good{}. The result follows then from Lemma \ref{lemma:filtration} and Proposition \ref{prop:sequentialhmtpycolimit}. \end{proof} ... ...
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