Commit 9e598626 by Leonard Guetta

### je vais bouffer

parent 8e96b687
 ... ... @@ -28,7 +28,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd} \] of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$. Composition is given by concatenation of chains. of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$ and is denoted by $\ell(f)$. Composition is given by concatenation of chains. \end{paragr} \begin{lemma} A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that ... ... @@ -89,7 +89,10 @@ In this section, we review some homotopical results concerning free ($1$-)catego X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd} \] of arrows of $L(G)$ such that the length of each $f_i$ is \emph{at most} $k$. In particular, we have of arrows of $L(G)$ such that $\sum_{1 \leq i \leq n}\ell(f_i) \leq n.$ In particular, we have $N^1(G)=i_!(G)$ ... ... @@ -113,15 +116,41 @@ In this section, we review some homotopical results concerning free ($1$-)catego \begin{proof} Let $A_{k+1}=\mathrm{Im}(\partial_0)\cup\mathrm{Im}(\partial_{k+1})$ be the union of the first and last face of the standard $(k+1)$-simplex $\Delta_{k+1}$. Notice that the canonical inclusion $A_{k+1} \hookedrightarrow \Delta_{k+1} A_{k+1} \hookrightarrow \Delta_{k+1}$ is a trivial cofibration. Let $I_{k+1}$ be the set of chains $\begin{tikzcd} f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1} \ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1} \end{tikzcd}$ of arrows of $L(G)$ such that for every $1 \leq i \leq k+1$ $\ell(f_i)=1,$ is a trivial cofibration. Now let $I_{k+1}$ be the set of chains i.e.\ each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$, we define a morphism $\varphi_f : A_{k+1} \to N^{k}(G)$ in the following fashion: \begin{itemize} \item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of $N^{k}(G)$ $\begin{tikzcd} X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k} \ar[r,"f_{k+1}"]& X_{k+1}, \end{tikzcd}$ \item[-] $\varphi_{f}\vert_{\mathrm{Im}(\partial_{k+1})}$ is the $k$-simplex of $N^{k}(G)$ $\begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1} \ar[r,"f_k"]& X_{k}. \end{tikzcd}$ \end{itemize} Now, we have a cocartesian square $\begin{tikzcd} f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\ \displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G), \end{tikzcd}$ where each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$ which proves that the right vertical arrow is a trivial cofibration. \end{proof} From this lemma, we deduce the following propositon. \begin{proposition} ... ... @@ -283,3 +312,39 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \end{remark} \todo{Préciser que si A=B dans l'exemple précédent ça ne marche pas ?} \section{Preliminaries : Bisimplicial sets} \begin{paragr} A \emph{bisimplicial set} is a presheaf over the category $\Delta \times \Delta$, $(\Delta \times \Delta)^{op} \to \Set.$ The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$. Let $X$ be a bisimplicial set. In a similar fashion as for simplicial sets, we use the notation $X_{n,m} := X([n],[m])$ for the image by $X$ of the object $([n],[m])$ of $\Delta\times \Delta$. Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set \begin{align*} X_{n,\bullet} : \Delta^{op} &\to \Set \\ [m] &\mapsto X_{n,m}. \end{align*} Similarly, if we fix the second variable to $n$, we obtain a simplicial \begin{align*} X_{\bullet,n} : \Delta^{op} &\to \Set \\ [m] &\mapsto X_{m,n}. \end{align*} The correspondances $n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}$ actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two currying'' operations $\Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{op},\Psh{\Delta}),$ which are isomorphisms of categories. In other words, the category of bisimplicial sets can be identified with the category of functors $\underline{\Hom}(\Delta^{op},\Psh{\Delta})$ in two canonical ways. \end{paragr} \begin{paragr} \end{paragr}
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