Commit 70c44319 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

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 $k<l\leq n$ 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.
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}
For any $n>0$, 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 $k<l\leq n$, 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 every $l$-cell $x$, we have
\[
\1^l_{t_k(x)}\comp_k x =x= x \comp_k\1^l_{s_k(x)},
\]
\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
\[
((x \comp_k x')\comp_l (y \comp_k y'))=((x \comp_l y)\comp_k (x' \comp_l y')).
\]
\end{enumerate}
We will use the same letter to denote an $n$-category and its underlying $n$-magma. Let $C$ and $C'$ be $n$-categories, a \emph{morphism of $n$-categories} (or $n$-functor) $f : C \to C'$ is simply a morphism of $n$-magmas. We denote by $n\Cat$ the category of $n$-categories and morphisms of $n$-categories.
\end{paragr}
\begin{paragr}
Once again, for any $n>0$ 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}
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment