Commit 2cdaa8c9 by Leonard Guetta

I have got to finish Lemmas 5.2.1 and 5.2.2

parent 8e07074f
 ... ... @@ -16,7 +16,7 @@ Recall that for any $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ the ca %% \begin{paragr} %% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason weak equivalence. In particular, we have the following lemma. %% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason equivalence. In particular, we have the following lemma. %% \end{paragr} %% \begin{lemma}\label{lemma:hmlgycontractible} ... ... @@ -74,22 +74,46 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}. \end{proof} \section{Homology of globes and spheres} \begin{paragr} \todo{Rappeler définitions par récurrence des globes et sphères.} \end{paragr} \begin{lemma}\label{lemma:globescontractible} For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is contractible. For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible. \end{lemma} \begin{proof} \todo{À écrire} Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}-cells for every $k$ such that $0\leq k  ... ... @@ -1012,12 +1012,12 @@ Proposition \ref{prop:freeiscofibrant} to show that there exists a free$n$\nbd{}category$C'$such that$\tau^{i}_{\leq n}(C')=C$. Let us write$\Sigma_k$for the$k$\nbd{}base of$C$with$0\leq k\leq n-1$and let$C'$be the free$\oo$\nbd{}category such that: \begin{itemize} \begin{itemize}[label=-] \item the$k$\nbd{}base of$C'$is$\Sigma_k$for every$0 \leq k \leq n-1$, \item the$n$\nbd{}base of$C'$is the set$C_n$, \item the$(n+1)$\nbd{}base of$C'$is the set $\{(x,y)\vert x \text{ and } y \text{ are parallel } n \text{-cells of } C\}, \{(x,y)\,\vert\, x \text{ and } y \text{ are parallel } n \text{-cells of } C\},$ the source (resp.\ target) of$(x,y)$being$x$(resp.\$y$), \item the$k$\nbd{}base of$C'$is empty for$k > n$(i.e.\$C'$is an$(n+1)$\nbd{}category). ... ... @@ -1025,7 +1025,7 @@ let$C'$be the free$\oo$\nbd{}category such that: (For such recursive constructions of free$\oo$\nbd{}categories, see Section \ref{section:freeoocataspolygraph}, and in particular Proposition \ref{prop:freeonpolygraph}.) We leave it to the reader to verify for himself that indeed$\tau^{i}_{\leq n}(C')=C$. We invite the reader to verify for himself that indeed$\tau^{i}_{\leq n}(C')=C$. \end{proof} \begin{example} Every (small) category is cofibrant for the folk model structure on$\Cat$. ... ... @@ -1220,7 +1220,7 @@ In the following lemma,$n\Cat$is equipped with the folk model structure and$\ The functor $\lambda_{\leq n} : n\Cat \to \Ch^{\leq n}$ is left Quillen. \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 $n\Cat$. Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$ as in the second part of such 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 $\Ch^{\leq n}$ (which we know exist by the second part of Proposition \ref{prop:fmsncat}). 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 $... ... No preview for this file type  ... ... @@ -246,27 +246,28 @@ A \emph{morphism of \oo-magmas} f : X \to Y is a morphism of underlying \oo the canonically associated \oo-functor. \end{paragr} \begin{paragr}\label{paragr:inclusionsphereglobe} For n \in \mathbb{N}, the n-sphere \sS^n is the n-category such for every k\leq n, it has exactly two parallel k-cells. In other words, we have For n \in \mathbb{N}, the n-sphere \sS_n is the n-category such for every k\leq n, it has exactly two parallel k-cells. In other words, we have \[ \sS^{n}=\sk_n(\sD_{n+1}), \sS_{n}=\sk_n(\sD_{n+1}),$ and in particular, we have a canonical inclusion functor $i_{n+1} : \sS^{n} \to \sD_{n+1}. i_{n+1} : \sS_{n} \to \sD_{n+1}.$ It is also customary to define $\sS^{-1}$ to be the empty $\oo$-category and $i_{-1}$ to be the unique $\oo$-functor It is also customary to define $\sS_{-1}$ to be the empty $\oo$-category and $i_{-1}$ to be the unique $\oo$-functor $\emptyset \to \sD_0.$ With this definition of $\sS^{-1}$, we can also give a recursive definition of the $n$-sphere for $n\geq 0$ as the following amalgamated sum Notice that for every $n\geq 0$, the following commutative square $\begin{tikzcd} \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d]\\ \sD_n \ar[r] & \sS_{n}. \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d,"j_n^+"]\\ \sD_n \ar[r,"j_n^-"'] & \sS_{n}, % \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}$ The two anonymous arrows of the previous square are representing each one of the two parallel $n$-cells of $\sS_n$. where we wrote $j_n^+$ (resp.\ $j_n^-$) for the morphism $\langle \trgt(e_{n+1}) \rangle : \sD_n \to \sS_n$ (resp.\ $\langle \src(e_{n+1}) \rangle : \sD_n \to \sS_n$), is cocartesian. % The two anonymous arrows of the previous square are representing each one of the two parallel $n$-cells of $\sS_n$. Here are some pictures of the $n$-spheres in low dimension: $... ... @@ -291,7 +292,7 @@ A \emph{morphism of \oo-magmas} f : X \to Y is a morphism of underlying \oo$ the canonically associated $\oo$-functor. For example, the $\oo$-functor $i_n$ is nothing but $\langle \src(e_{n+1}),\trgt(e_{n+1}) \rangle : \sS^{n} \to \sD_{n+1}. \langle \src(e_{n+1}),\trgt(e_{n+1}) \rangle : \sS_{n} \to \sD_{n+1}.$ \end{paragr} \section{Free $\oo$-categories} ... ... @@ -299,7 +300,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo Let$C$be an$\oo$-category and$n \geq 0$. A subset$E \subseteq C_n$of the$n$-cells of$C$is an \emph{$n$-basis of$C$} if the commutative square $\begin{tikzcd}[column sep=huge, row sep=huge] \displaystyle \coprod_{ x \in E}\ar[d,"\displaystyle\coprod_{x \in E} i_n"'] \sS^{n-1} \ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in E}}"] & \sk_{n-1}(C) \ar[d,hook] \\ \displaystyle \coprod_{ x \in E}\ar[d,"\displaystyle\coprod_{x \in E} i_n"'] \sS_{n-1} \ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in E}}"] & \sk_{n-1}(C) \ar[d,hook] \\ \displaystyle \coprod_{x \in E} \sD_n \ar[r,"\langle x \rangle_{x \in E}"] & \sk_{n}(C) \end{tikzcd}$ ... ... @@ -586,7 +587,7 @@ Furthermore, this function satisfies the condition On the other hand, any$\E=(D,\Sigma,\sigma,\tau)$, cellular extension of an$n$-category$D$, yields an$(n+1)$-category$\E^*$defined as the following amalgamated sum: \label{squarefreecext} \begin{tikzcd}[column sep=huge, row sep=huge] \displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & D \ar[d] \\ \displaystyle\coprod_{x \in \Sigma}\sS_n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & D \ar[d] \\ \displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\E^* \ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"] \end{tikzcd}. ... ... @@ -643,10 +644,10 @@ We can now prove the following proposition, which is the key result of this sect \end{proposition} \begin{proof} Notice first that since the map$i_{n+1} : \sS^n \to \sD_{n+1}$is nothing but the canonical inclusion$\sk_{n}(\sD_{n+1}) \to \sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from square \eqref{squarefreecext} and the fact the the skeleton functors preserve colimits that$C$is canonically isomorphic to$\sk_n(\E^*)$and that the map$C \to \E^*$can be identified with the canonical inclusion$\sk_n(\E^*) \to \sk_{n+1}(\E^*)=\E^*$. Hence, cocartesian square \eqref{squarefreecext} can be identified with Notice first that since the map$i_{n+1} : \sS_n \to \sD_{n+1}$is nothing but the canonical inclusion$\sk_{n}(\sD_{n+1}) \to \sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from square \eqref{squarefreecext} and the fact the the skeleton functors preserve colimits that$C$is canonically isomorphic to$\sk_n(\E^*)$and that the map$C \to \E^*$can be identified with the canonical inclusion$\sk_n(\E^*) \to \sk_{n+1}(\E^*)=\E^*\$. Hence, cocartesian square \eqref{squarefreecext} can be identified with \[ \begin{tikzcd}[column sep=huge, row sep=huge] \displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\ \displaystyle\coprod_{x \in \Sigma}\sS_n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\ \displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*). \ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"] \end{tikzcd} ... ...
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