Commit 9e598626 authored by Leonard Guetta's avatar Leonard Guetta
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je vais bouffer

parent 8e96b687
...@@ -28,7 +28,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego ...@@ -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} 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} \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} \end{paragr}
\begin{lemma} \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 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 ...@@ -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} 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} \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
\[ \[
N^1(G)=i_!(G) N^1(G)=i_!(G)
\] \]
...@@ -113,15 +116,41 @@ In this section, we review some homotopical results concerning free ($1$-)catego ...@@ -113,15 +116,41 @@ In this section, we review some homotopical results concerning free ($1$-)catego
\begin{proof} \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 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
\[ \[
\begin{tikzcd} \begin{tikzcd}
f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\
\displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G),
\end{tikzcd} \end{tikzcd}
\] \]
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} \end{proof}
From this lemma, we deduce the following propositon. From this lemma, we deduce the following propositon.
\begin{proposition} \begin{proposition}
...@@ -283,3 +312,39 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp ...@@ -283,3 +312,39 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\end{remark} \end{remark}
\todo{Préciser que si A=B dans l'exemple précédent ça ne marche pas ?} \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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