Commit 70c44319 by Leonard Guetta

### Ok, I gave the definition of oo-categories

parent 30b65d41
 ... ... @@ -51,6 +51,7 @@ \newcommand{\nGrph}{n \mathbf{Grph}} \newcommand{\ooMag}{\mathbf{\oo Mag}} \newcommand{\nMag}{n \mathbf{Mag}} \newcommand{\Mag}{\mathbf{Mag}} % compositions and units \def\1^#1_#2{1^{(#1)}_{#2}} ... ...
 ... ... @@ -49,7 +49,7 @@ \] That is, elements of $C_l\underset{C_k}{\times}C_l$ are pairs $(x,y)$ of $l$-cells such that $s_k(x)=t_k(y)$. We say that two $l$-cells $x$ and $y$ are \emph{$k$-composable} if the pair $(x,y)$ belongs to $C_l\times_{C_k}C_l$. Let $C$ and $C'$ be two $\oo$-graphs. A \emph{morphism of $\oo$-graphs} $f : C \to C'$ is a sequence $(f_k : C_k \to D_k)_{0 \leq k \leq n}$ of maps such that for every $0< k \leq n$, the squares Let $C$ and $C'$ be two $n$-graphs. A \emph{morphism of $n$-graphs} $f : C \to C'$ is a sequence of maps $(f_k : C_k \to D_k)_{0 \leq k \leq n}$ such that for every $0< k \leq n$, the squares $\begin{tikzcd} C_k \ar[d,"s"] \ar[r,"f_k"]&C'_k \ar[d,"s"] \\ ... ... @@ -71,8 +71,148 @@ \[ \nGrph \to (n\shortminus 1)\Grph$ that simply forgets the $n$-cells. We define the category $\oo\Grph$ as the limit of the diagram that simply forgets the $n$-cells. We define the category $\oo\Grph$ of $\omega$-graphs as the limit of the diagram $\cdots \to \nGrph \to (n\shortminus 1)\Grph \to \cdots \to 0\Grph \to 1\Grph. \cdots \to \nGrph \to (n\shortminus 1)\Grph \to \cdots \to 1\Grph \to 0\Grph.$ \end{paragr} \begin{remark} More concretely, the definitions of $\omega$-graph and morphism of $\omega$-graphs are the same as the definitions of $n$-graph and morphisms of $n$-graphs with $n$ finite but with the condition $\leq n$'' dropped everywhere. For example, an $\omega$-graph has an infinite sequence of cells $(C_k)_{k\in \mathbb{N}}$. \remtt{Est-ce que je devrais plus détailler la définition de $\oo$-graphe ?} \end{remark} \begin{paragr} Let $n \in \mathbb{N}$. An \emph{$n$-magma} consists of: \begin{itemize} \item[-] an $n$-graph $C$, \item[-] maps \begin{aligned} (\shortminus)\underset{k}{\ast}(\shortminus) : C_l\underset{C_k}{\times}C_l &\to C_l \\ (x,y) &\mapsto x\underset{k}{\ast}y \end{aligned} for all $l,k \in \mathbb{N}$ with $k < l \leq n$, \item[-] maps \begin{aligned} 1_{(\shortminus)} : C_k &\to C_{k+1}\\ x &\mapsto 1_x \end{aligned} for every $k \in \mathbb{N}$ with $k\leq n$, \end{itemize} subject to the following axioms: \begin{itemize} \item[-] for all $k,l \in \mathbb{N}$ with $k0$, there is an obvious truncation'' functor $n\Mag \to (n\shortminus 1)\Mag$ that simply forgets the $n$-cells. We define the category $\oo\Mag$ of $\omega$-magmas as the limit of the diagram $\cdots \to n\Mag \to (n\shortminus 1)\Mag \to \cdots \to 1\Mag \to 0\Mag.$ Moreover, for any $n\in \mathbb{N}$, there is a canonical forgetful functor $n\Mag \to n\Grph$ such that when $n>0$ the square $\begin{tikzcd} n\Mag\ar[r] \ar[d] & n\Grph \ar[d]\\ (n \shortminus 1)\Mag \ar[r] & (n \shortminus 1)\Grph \end{tikzcd}$ is commutative. By universal property, this yields a canonical forgetful functor $\oo\Mag \to \oo\Grph.$ \end{paragr} \begin{paragr} Let $n \in \mathbb{N}$. An \emph{$n$-category} $C$ is an $n$-magma such that the following axioms are satisfied: \begin{enumerate} \item for all $k,l \in \mathbb{N}$ with $k0$ there is a canonical truncation'' functor $n\Cat \to (n \shortminus 1)\Cat$ that simply forgets the $n$-cells. We define the category $\oo\Cat$ of $\oo$-categories as the limit of the diagram $\cdots \to n\Cat \to (n\shortminus 1)\Cat \to \cdots \to 1\Cat \to 0\Cat.$ \end{paragr}
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