Commit 761cc007 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

lunch breack

parent 7a83a46d
...@@ -417,18 +417,20 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph, ...@@ -417,18 +417,20 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph,
The \emph{length} of a word $w$, denoted by $\mathcal{L}(w)$, is the number of symbols that appear in $w$. The \emph{length} of a word $w$, denoted by $\mathcal{L}(w)$, is the number of symbols that appear in $w$.
\end{paragr} \end{paragr}
\begin{paragr} \begin{paragr}
We now recursively define the set $\Sigma^{+} \subseteq \W[\Sigma]$ of \emph{well formed words} (or \emph{terms}) on this alphabet together with maps $\sigma,\tau : \Sigma^{+} \to C_{n-1}$ that satisfy the globular conditions: We now recursively define the set $\Sigma^{+} \subseteq \W[\Sigma]$ of \emph{well formed words} (or \emph{terms}) on this alphabet together with maps $s,t : \Sigma^{+} \to C_{n-1}$ that satisfy the globular conditions:
\begin{itemize} \begin{itemize}
\item[-] $(\hat{x}) \in \T[E]$ with $\sigma((\hat{x}))=\sigma(\alpha)$ and $t((\hat{x}))=\tau((\hat{x}))$ for each $x \in \Sigma$, \item[-] $(\hat{x}) \in \Sigma^{+}$ with
%\item[-] $(\ii_{x}) \in \T[E]$ with $s((\ii_x))=t((\ii_x))=x$ for each $x \in C_n$, \[s((\hat{x}))=s(x) \text{ and }t((\hat{x}))=t(x)\]
%\item[-] $ (v \ast_n w) \in \T[E]$ with $s((v \ast_n w))=s(w)$ and $t((v \ast_n w))=t(v)$ for $v,w \in \T[E]$ such that $s(v)=t(w)$, for each $x \in \Sigma$,
% \item[-] $(v \ast_k w) \in \T[E]$ with \[s((v \ast_k w)) = s(v) \ast_k s(w)\] and \[t((v \ast_k w))=t(v)\ast_k t(w)\] for $v, w \in \T[E]$ and $0 \leq k < n$, such that $s^k(s(v))=t^k(t(w))$. \item[-] $(\ii_{x}) \in \Sigma^{+}$ with $s((\ii_x))=t((\ii_x))=x$ for each $x \in C_n$,
\item[-] $ (v \comp_n w) \in \Sigma^{+}$ with $s((v \comp_n w))=s(w)$ and $t((v \comp_n w))=t(v)$ for $v,w \in \Sigma^{+}$ such that $s(v)=t(w)$,
\item[-] $(v \hat{\comp_k} w) \in \Sigma^{+}$ with \[s((v \hat{\comp_k} w)) = s(v) \comp_k s(w)\] and \[t((v \hat{\comp_k} w))=t(v)\comp_k t(w)\] for $v, w \in \Sigma^{+}$ and $0 \leq k < n$, such that $s_k(s(v))=t_k(t(w))$.
\end{itemize} \end{itemize}
%As usual, if $w \in \T[E]$ we often write $w : x \to y$ to say that $s^n(w)=x$ and $t^n(w)=y$. %As usual, if $w \in \Sigma^{+}$ we often write $w : x \to y$ to say that $s_n(w)=x$ and $t_n(w)=y$.
We define $s^k , t^k: \T[E] \to C_k$ 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 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
\[s(v)=s(w) \text{ and }t(v)=t(w)\] \[s(v)=s(w) \text{ and }t(v)=t(w)\]
and we say that they are \emph{$k$-composable} for a $k\leq n$ if and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\[s^k(v)=t^k(w).\] \[s_k(v)=t_k(w).\]
\end{paragr} \end{paragr}
\section{$\oo$-categories} \section{$\oo$-categories}
\begin{paragr} \begin{paragr}
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