Commit 30b65d41 by Leonard Guetta

### I started doing what I said in the previous commit. I also added a file...

I started doing what I said in the previous commit. I also added a file recyclebin.tex  that contains the former version omegacat.tex
parent c3df655e
 ... ... @@ -47,6 +47,7 @@ \newcommand{\ooCat}{\mathbf{\oo Cat}} \newcommand{\nCat}{n \mathbf{Cat}} \newcommand{\ooGrph}{\mathbf{\oo Grph}} \newcommand{\Grph}{\mathbf{Grph}} \newcommand{\nGrph}{n \mathbf{Grph}} \newcommand{\ooMag}{\mathbf{\oo Mag}} \newcommand{\nMag}{n \mathbf{Mag}} ... ...
 \chapter{$\oo$-Category theory} \section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories} \begin{paragr}\label{paragr:defoograph} An \emph{$\oo$-graph} $C$ consists of a sequence $(C_k)_{k \in \mathbb{N}}$ of sets together with maps \begin{paragr}\label{pargr:defngraph} Let $n \in \mathbb{N}$. An \emph{$n$-graph} $C$ consists of a finite sequence $(C_k)_{0\leq k \leq n}$ of sets together with maps $\begin{tikzcd} C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k} \end{tikzcd}$ for every $k > 0$, subject to the \emph{globular identities}: for every $0 < k \leq n$, subject to the \emph{globular identities}: \begin{equation*} \left\{ \begin{aligned} ... ... @@ -16,10 +16,16 @@ \end{aligned} \right. \end{equation*} Elements of $C_k$ are called \emph{$k$-cells} or \emph{cells of dimension $k$}. For example, a $0$-graph is just a set and a $1$-graph is an ordinary graph: $\begin{tikzcd} C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_1. \end{tikzcd}$ Elements of $C_k$ are called \emph{$k$-cells} or \emph{cells of dimension $k$}. For $x$ a $k$-cell with $k>0$, $s(x)$ is the \emph{source} of $x$ and $t(x)$ is the \emph{target} of $x$. More generally for all $k0$ and $s(x)=s(y) \text{ and } t(x)=t(y).$ For all $k0$, the squares 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 $\begin{tikzcd} C_k \ar[d,"s"] \ar[r,"f_k"]&C'_k \ar[d,"s"] \\ ... ... @@ -60,111 +64,15 @@ are commutative. For a k-cell x, we will often write f(x) instead of f_k(x). We denote by \ooGrph the category of \oo-graphs and morphisms of \oo-graphs. \end{paragr} We denote by \nGrph the category of n-graphs and morphisms of n-graphs. \end{paragr} \begin{paragr} For n \in \mathbb{N}, the notion of \emph{n-graph} is defined similarly, only this time there is only a finite sequence (C_k)_{0 \leq k \leq n} of cells. For example, a 0-graph is just a set and a 1-graph is an ordinary graph For any n>0, there is an obvious truncation'' functor \[ \begin{tikzcd} C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_1. \end{tikzcd} \nGrph \to (n\shortminus 1)\Grph$ that simply forgets the $n$-cells. We define the category $\oo\Grph$ as the limit of the diagram $\cdots \to \nGrph \to (n\shortminus 1)\Grph \to \cdots \to 0\Grph \to 1\Grph.$ The definition of morphism of $n$-graphs is the same as for $\omega$-graphs, only this time there is only a finite sequence $(f_k : C_k \to C'_k)_{0 \leq k \leq n}$ of maps. We denote by $\nGrph$ the category of $n$-graphs and morphisms of $n$-graphs. \end{paragr} \begin{paragr} Let $n \in \nbar$. 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 $k < l \leq n$,\footnote{Note that if $n=\omega$, then $l recyclebin.tex 0 → 100644  \begin{paragr}\label{paragr:defoograph} An \emph{$\oo$-graph}$C$consists of a sequence$(C_k)_{k \in \mathbb{N}}$of sets together with maps $\begin{tikzcd} C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k} \end{tikzcd}$ for every$k > 0, subject to the \emph{globular identities}: \begin{equation*} \left\{ \begin{aligned} s \circ s &= s \circ t, \\ t \circ t &= t \circ s. \end{aligned} \right. \end{equation*} Elements ofC_k$are called \emph{$k$-cells} or \emph{cells of dimension$k$}. For$x$a$k$-cell with$k>0$,$s(x)$is the \emph{source} of$x$and$t(x)$is the \emph{target} of$x$. More generally for all$k0$and $s(x)=s(y) \text{ and } t(x)=t(y).$ For all$k0$, the squares $\begin{tikzcd} C_k \ar[d,"s"] \ar[r,"f_k"]&C'_k \ar[d,"s"] \\ C_{k-1} \ar[r,"f_{k-1}"] & C'_{k-1} \end{tikzcd} \quad \begin{tikzcd} C_k \ar[d,"t"] \ar[r,"f_k"]&C'_k \ar[d,"t"] \\ C_{k-1} \ar[r,"f_{k-1}"] & C'_{k-1} \end{tikzcd}$ are commutative. For a$k$-cell$x$, we will often write$f(x)$instead of$f_k(x)$. We denote by$\ooGrph$the category of$\oo$-graphs and morphisms of$\oo$-graphs. \end{paragr} \begin{paragr} For$n \in \mathbb{N}$, the notion of \emph{$n$-graph} is defined similarly, only this time there is only a finite sequence$(C_k)_{0 \leq k \leq n}$of cells. For example, a$0$-graph is just a set and a$1$-graph is an ordinary graph $\begin{tikzcd} C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_1. \end{tikzcd}$ The definition of morphism of$n$-graphs is the same as for$\omega$-graphs, only this time there is only a finite sequence$(f_k : C_k \to C'_k)_{0 \leq k \leq n}$of maps. We denote by$\nGrph$the category of$n$-graphs and morphisms of$n$-graphs. \end{paragr} \begin{paragr} Let$n \in \nbar$. An \emph{$n$-magma} consists of: \begin{itemize} \item[-] an$n$-graph$C, \item[-] maps \begin{aligned} (\shortminus)\underset{k}{\ast}(\shortminus) : C_l\underset{C_k}{\times}C_l &\to C_l \\ (x,y) &\mapsto x\underset{k}{\ast}y \end{aligned} for alll,k \in \mathbb{N}$with$k < l \leq n$,\footnote{Note that if$n=\omega$, then$l
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