Commit 842fb754 authored by Leonard Guetta's avatar Leonard Guetta
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lunck break

parent 0857c269
......@@ -2,6 +2,7 @@
\usepackage{mystyle}
\begin{document}
\chapter{Yoga of $\oo$-Categories : OLD}
\section{$n$-graphs, $n$-magmas and $n$-categories}
......@@ -514,3 +515,4 @@ and we say that they are \emph{$k$-composable} for a $k< n$ if
\todo{...}
test
\end{proof}
\end{document}
\chapter{Yoga of $\oo$-categories}
\section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories}
\begin{paragr}\label{paragr:defngraph}
An $\oo$-graph $C$ consists of an infinite sequence of sets $(X_k)_{k \in \mathbb{N}}$ together with maps
An \emph{$\oo$-graph} $X$ consists of an infinite sequence of sets $(X_n)_{n \in \mathbb{N}}$ together with maps
\[ \begin{tikzcd}
X_{k} &\ar[l,"\src",shift left] \ar[l,"\trgt"',shift right] X_{k+1}
X_{n} &\ar[l,"\src",shift left] \ar[l,"\trgt"',shift right] X_{n+1}
\end{tikzcd}
\]
for every $k \in \mathbb{N}$, subject to the \emph{globular identities}:
for every $n \in \mathbb{N}$, subject to the \emph{globular identities}:
\begin{equation*}
\left\{
\begin{aligned}
......@@ -15,13 +15,34 @@
\end{aligned}
\right.
\end{equation*}
Elements of $X_k$ are called \emph{$k$-cells} or \emph{cells of dimension $k$}. For $x$ a $k$-cell with $k>0$, $\src(x)$ is the \emph{source} of $x$ and $\trgt(x)$ is the \emph{target} of $x$. More generally, for $0\leq i <k$, we define maps $\src_i : X_k \to X_i$ and $\trgt_i : X_k \to X_i$ as
Elements of $X_n$ are called \emph{$n$-cells} or \emph{cells of dimension $n$}. For $n=0$, elements of $X_0$ are also called \emph{objects}. For $x$ a $n$-cell with $n>0$, $\src(x)$ is the \emph{source} of $x$ and $\trgt(x)$ is the \emph{target} of $x$. More generally, for $0\leq k <n$, we define maps $\src_k : X_n \to X_k$ and $\trgt_k : X_n \to X_k$ as
\[
\src_i = \underbrace{\src\circ \dots \circ \src}_{k-i \text{ times}}
\src_k = \underbrace{\src\circ \dots \circ \src}_{n-k \text{ times}}
\]
and
\[
\trgt_i = \underbrace{\trgt\circ \dots \circ \trgt}_{k-i \text{ times}}.
\trgt_k = \underbrace{\trgt\circ \dots \circ \trgt}_{n-k \text{ times}}.
\]
For a $k$-cell $x$, the $i$-cells $\src_i(x)$ and $\trgt_i(x)$ are respectively the \emph{$i$-source} and the \emph{$i$-target} of $x$.
For a $n$-cell $x$, the $k$-cells $\src_k(x)$ and $\trgt_k(x)$ are respectively the \emph{$k$-source} and the \emph{$k$-target} of $x$.
Two $n$-cells $x$ and $y$ are \emph{parallel} if
\[
n=0
\]
or
\[
n>0 \text{ and }\src(x)=\src(y) \text{ and } \trgt(x)=\trgt(y).
\]
Let $0 \leq k <n$. Two $n$-cells $x$ and $y$ are \emph{$k$-composable} if
\[
\src_k(x)=\trgt_k(y).
\]
Note that the expression ``$x$ and $y$ are $k$-composable'' is \emph{not} symetric in $x$ and $y$ and we \emph{should} rather speak of a ``$k$-composable pair $(x,y)$'', although we won't always do it. The set of pairs of $k$-composable $n$-cells is denoted by $C_n\underset{C_k}{\times}C_n$, and is characterized by the following fibred product
\[
\begin{tikzcd}
C_n\underset{C_k}{\times}C_n \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &C_n \ar[d,"\trgt_k"]\\
C_n \ar[r,"\src_k"] & C_k.
\end{tikzcd}
\]
\end{paragr}
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