Commit c3df655e authored by Leonard Guetta's avatar Leonard Guetta
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I started writing up the definitions of oo-graphs, oo-magmas and...

I started writing up the definitions of oo-graphs, oo-magmas and oo-categories. I would like to try another approach where I start with the finite case (n-graphs, n-magmas, etc.) and define the oo counterparts as projective limits
parent b5de6425
......@@ -3,6 +3,9 @@
% Layout
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{showlabels}
% Maths packages
......@@ -42,13 +45,22 @@
\newcommand{\oo}{\omega}
\newcommand{\Cat}{\mathbf{Cat}}
\newcommand{\ooCat}{\mathbf{\oo Cat}}
\newcommand{\nCat}{\mathbf{n Cat}}
\newcommand{\nCat}{n \mathbf{Cat}}
\newcommand{\ooGrph}{\mathbf{\oo Grph}}
\newcommand{\nGrph}{n \mathbf{Grph}}
\newcommand{\ooMag}{\mathbf{\oo Mag}}
\newcommand{\nMag}{n \mathbf{Mag}}
% Source and target
\newcommand{\s}[1]{s^{(#1)}}
\newcommand{\sk}{\s{k}}
\newcommand{\t}[1]{t^{(#1)}}
\newcommand{\tk}{\t{k}}
% compositions and units
\def\1^#1_#2{1^{(#1)}_{#2}}
\def\comp_#1{\underset{#1}{\ast}}
% ad hoc
\newcommand{\nbar}{\mathbb{N}\cup\{ \oo \}}
% Maths
\DeclareMathSymbol{\shortminus}{\mathbin}{AMSa}{"39} %For short minus signs
% commentaires
\newcommand\remtt[1]{\texttt{[#1]}}
\newcommand\todo[1]{\remtt{TODO : #1}}
\chapter{$\oo$-Category theory}
\section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories}
\begin{paragr}
\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 $k > 0$, satisfying the \emph{globular identities}:
for every $k > 0$, subject to the \emph{globular identities}:
\begin{equation*}
\left\{
\begin{aligned}
......@@ -16,13 +16,155 @@
\end{aligned}
\right.
\end{equation*}
Elements of $C_k$ are called \emph{$k$-cells}.
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 $x$.
%Similarly, for $n \in \mathbb{N}$, an \emph{$n$-graph} $C$ is 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 $0 <k < n$, satisfying the same globular identities.
More generally for all $k<n \in \mathbb{N}$, we define maps $s_k,t_k : C_n \to C_k$ as
\[
s_k = \underbrace{s\circ \dots \circ s}_{n-k \text{ times}}
\]
and
\[
t_k = \underbrace{t\circ \dots \circ t}_{n-k \text{ times}}.
\]
For an $n$-cell $x$, $s_k(x)$ is the \emph{$k$-source} of $x$ and $t_k(x)$ the \emph{$k$-target} of $x$.
Two $k$-cells $x$ and $y$ are \emph{parallel} if $k=0$ or $k>0$ and
\[
s(x)=s(y) \text{ and } t(x)=t(y).
\]
For all $k<n \in \mathbb{N}$, we define the set $C_n\underset{C_k}{\times}C_n$ as the following fibred product
\[
\begin{tikzcd}
C_n\underset{C_k}{\times}C_n \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &C_n \ar[d,"t_k"]\\
C_n \ar[r,"s_k"] & C_k.
\end{tikzcd}
\]
That is, elements of $C_n\underset{C_k}{\times}C_n$ are pairs $(x,y)$ of $n$-cells such that $s_k(x)=t_k(y)$. We say that two $n$-cells $x$ and $y$ are \emph{$k$-composable} if the pair $(x,y)$ belongs to $C_n\times_{C_k}C_n$.
\end{paragr}
\begin{paragr}\label{paragr:defmoroograph}
Let $C$ and $C'$ be two $\oo$-graphs. A \emph{morphism of $\oo$-graphs} $f : C \to C'$ from $C$ to $C'$ is a sequence $(f_k : C_k \to D_k)_{k \in \mathbb{N}}$ of maps such that for every $k>0$, 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 all $l,k \in \mathbb{N}$ with $k < l \leq n$,\footnote{Note that if $n=\omega$, then $l<n$ because we supposed that $l \in \mathbb{N}$.}
\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 $k<l\leq n$ and every $k$-composable $l$-cells $x$ and $y$,
\[
s(x\underset{k}{\ast} y) =
\begin{cases}
s(y) &\text{ when }k=l-1,\\
s(x)\underset{k}{\ast} s(y) &\text{ otherwise,}
\end{cases}
\]
and
\[
t(x\underset{k}{\ast} y) =
\begin{cases}
t(x) & \text{ when }k=l-1,\\
t(x)\underset{k}{\ast} t(y) &\text{ otherwise.}
\end{cases}
\]
\item[-]for every $k \in \mathbb{N}$ with $k\leq n$ and every $k$-cell,
\[
s(1_x)=t(1_x)=x.
\]
\end{itemize}
We will use the same letter to denote an $n$-magma and its underlying $n$-graph.
For two $k$-composable $l$-cells $x$ and $y$, we refer to $x\ast_ky$ as the \emph{$k$-composition} of $x$ and $y$.
For a $k$-cell $x$, we refer to $1_{x}$ as the \emph{unit on $x$}.
\remtt{Je n'aime pas trop les notations et définitions des unités itérées qui suivent.}
More generally, for any $l \in \mathbb{N}$ with $k < l\leq n$, we define $\1^{l}_{(\shortminus)} : C_k \to C_l$ as
\[
\1^{l}_{(\shortminus)} := \underbrace{1_{(\shortminus)} \circ \dots \circ 1_{(\shortminus)}}_{l-k \text{ times }} : C_k \to C_l.
\]
Let $x$ be a $k$-cell, $\1^l_x$ is the \emph{$l$-dimensional unit on $x$}, and for consistency, we also set
\[
\1^{k}_x := x.
\]
A cell is \emph{degenerate} if it is a unit on a strictly lower dimensional cell.
\end{paragr}
\begin{paragr}
Let $n \in \nbar$ and let $C$ and $C'$ be $n$-magmas. A \emph{morphism of $n$-magmas} $f : C \to C'$ is a morphism of $n$-graphs that is compatible with compositions and units. This means:
\begin{itemize}
\item[-]for all $k,l \in \mathbb{N}$ with $k<l\leq n$ and every $k$-composable $l$-cells $x$ and $y$,
\[
f(x\underset{k}{\ast}y)=f(x)\underset{k}{\ast}f(y),
\]
\item[-]for every $k \in \mathbb{N}$ with $k\leq n$ and every $k$-cell $x$,
\[
f(1_x)=1_{f(x)}.
\]
\end{itemize}
We denote by $\nMag$ the category of $n$-magmas and morphisms of $n$-magmas.
\end{paragr}
\begin{paragr}
Let $n \in \nbar$. 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 $k<l\leq n$, for all $k$-composable $l$-cells $x$ and $y$, we have
\[
1_{x\underset{k}{\ast}y}=1_{x}\underset{k}{\ast}1_{y},
\]
\item for all $k,l \in \mathbb{N}$ with $k<l\leq n$, for all $l$-cells $x, y$ and $z$ such that $x$ and $y$ are $k$-composable, and $y$ and $z$ are $k$-composable, we have
\[
(x\comp_{k}y)\comp_{k}z=x\comp_k(y\comp_kz),
\]
\item for all $k, l \in \mathbb{N}$, for all $n$-cells $x,x',y$ and $y'$ such that
\begin{itemize}
\item[-] $x$ and $y$ are $l$-composable, $x'$ and $y'$ are $l$-composable,
\item[-] $x$ and $x'$ are $k$-composable, $y$ and $y'$ are $k$-composable,
\end{itemize}
we have
\[
((x \ast^n_k x')\ast^n_l (y \ast^n_k y'))=((x \ast^n_l y)\ast^n_k (x' \ast^n_l y')).
\]
\end{enumerate}
\end{paragr}
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