### Fini pour ajd

parent 761cc007
 ... ... @@ -80,9 +80,12 @@ \newcommand{\CellExt}{\mathbf{CellExt}} % compositions and units \def\1^#1_#2{1^{(#1)}_{#2}} \def\1^#1_#2{1^{(#1)}_{#2}} % for iterated units \def\comp_#1{\underset{#1}{\ast}} \def\comp_#1{\mathbin{\underset{#1}{\ast}}} % for composition \def\fcomp_#1{\mathbin{\hat{\underset{#1}{\ast}}}} % formal composition % useful stuff ... ...
 ... ... @@ -404,33 +404,49 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph, We will now explicitely construct a left adjoint of $U$. In order to do that, we will successively construct left adjoints of $W$ and $V$. \end{paragr} \begin{paragr} Let $(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has: Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has: \begin{itemize} \item[-] a symbol $\hat{x}$ for each $x \in \Sigma$, \item[-] a symbol $\hat{\comp_k}$ for each $k  ... ... @@ -171,3 +171,24 @@ n\Cat \ar[r,"\tau",""{name=B,above},""{name=C,below}] & (n\shortminus 1)\Cat \ar[l,bend right,"\iota"',""{name=A, below}] \ar[l,bend left,"\kappa",""{name=D, above}] \ar[from=A,to=B,symbol=\dashv]\ar[from=C,to=D,symbol=\dashv]. \end{tikzcd} \] For every$k\in \mathbb{N}$such that$k < n$, we define the set$\Sigma^{+}\underset{C_k}{\times}\Sigma^{+}$as the following fibred product $\begin{tikzcd} \Sigma^{+}\underset{C_k}{\times}\Sigma^{+} \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &\Sigma^{+} \ar[d,"t_k"]\\ \Sigma^{+} \ar[r,"s_k"] & C_k. \end{tikzcd}$ That is, elements of$\Sigma^{+}\underset{C_k}{\times}\Sigma^{+}$are pairs$(x,y)$of well formed words such that$s_k(x)=t_k(y)$. We say that two well formed words$x$and$y$are \emph{$k$-composable} if the pair$(x,y)$belongs to$\Sigma^{+}\times_{C_k}\Sigma^{+}$. \end{paragr} We define$s_k , t_k: \Sigma^{+} \to C_k$as iterated source and target (with$s_n=s$and$t_n=t$for consistency). We say that two well formed words$v$and$w$are \emph{parallel} if $s(v)=s(w) \text{ and }t(v)=t(w).$ and we say that they are \emph{$k$-composable} for a$k\leq n$if $s_k(v)=t_k(w).$ It is straightforward to check that $\begin{tikzcd} C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] \cdots & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_{n \shortminus 1} & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] \Sigma^{+} \end{tikzcd}$ is an$n\$-graph and
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