Commit 3fe08a7b authored by Leonard Guetta's avatar Leonard Guetta
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Il faut que j'avance plus vite!

parent a29f27e9
\chapter{Yoga of $\oo$-Categories}
\section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories}
\section{$n$-graphs, $n$-magmas and $n$-categories}
\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
......@@ -65,21 +65,14 @@
For a $k$-cell $x$, we will often write $f(x)$ instead of $f_k(x)$.
We denote by $\nGrph$ the category of $n$-graphs and morphisms of $n$-graphs.
\end{paragr}
\begin{paragr}
For any $n>0$, there is an obvious ``truncation'' functor
\[
\nGrph \to (n\shortminus 1)\Grph
\]
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 1\Grph \to 0\Grph.
For any $n>0$, there is an obvious ``truncation'' functor
\[
\tau : n\Grph \to (n\shortminus 1)\Grph
\]
\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}
that simply forgets the $n$-cells. That is to say, for $C$ an $n$-graph, $\tau(C)$ is the $(n \shortminus 1)$-graph with $\tau(C)_k=C_k$ for every $0 \leq k \leq n\shortminus 1$.
\end{paragr}
\begin{paragr}
Let $n \in \mathbb{N}$. An \emph{$n$-magma} consists of:
\begin{itemize}
......@@ -153,32 +146,21 @@
\]
\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
For any $n>0$, there is an obvious ``truncation'' functor
\[
n\Mag \to (n\shortminus 1)\Mag
\tau : n\Mag \to (n\shortminus 1)\Mag
\]
that simply forgets the $n$-cells. We define the category $\oo\Mag$ of $\omega$-magmas as the limit of the diagram
that simply forgets the $n$-cells. Moreover, the square
\[
\cdots \to n\Mag \to (n\shortminus 1)\Mag \to \cdots \to 1\Mag \to 0\Mag.
\]
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{tikzcd}
n\Mag \ar[r,"\tau"]\ar[d] & (n\shortminus 1)\Mag \ar[d]\\
n\Grph \ar[r,"\tau"] & (n\shortminus 1)\Grph
\end{tikzcd}
\]
where the vertical arrows are the obvious forgetful functors, is commutative.
\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}
......@@ -206,6 +188,140 @@
\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}
\section{Generating cells}
\begin{paragr}
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\[
\begin{tikzcd}
n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Grph \ar[r] & (n \shortminus 1)\Grph.
\end{tikzcd}
\]
More concretely, an $n$-cellular extension can be encoded in the data of a quadruple $(\Sigma,C,s,t)$ where $\Sigma$ is a set, $C$ is a $(n\shortminus 1)$-category, $s$ and $t$ are maps
\[
s,t : \Sigma \to C_n
\]
such that
\[
\begin{tikzcd}
C_{n-1} & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] C_n & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] \Sigma
\end{tikzcd}
\]
satisfy the globular identities.
Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with extra $n$-cells that make it a $n$-graph.
A morphism of $n$-cellular extensions from $(\Sigma,C,s,t)$ to $(\Sigma',C',s',t')$ consists of a pair $(\varphi,f)$ where $f : C \to C'$ is a morphism of $(n\shortminus 1)\Cat$ and $\varphi : \Sigma \to \Sigma'$ is a map such that the squares
\[
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\text{ and }
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"t"] & \Sigma' \ar[d,"t'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\]
commute.
%From now on, we will denote such an object of $n\GCat$ by
%\[
%\begin{tikzcd}
% \Sigma \ar[r,shift left,"s"] \ar[r,shift right,"t"']& C
% \end{tikzcd}
%\]
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\[
\begin{tikzcd}
n\PCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Mag \ar[r] & (n \shortminus 1)\Mag.
\end{tikzcd}
\]
More concretely, objects of $n\PCat$ can be seen as $n$-magmas such that if we forget their $n$-cells then they satisfy the axioms of $(n\shortminus 1)$-category. The left vertical arrow of the previous square is easily seen to be full, and we will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$.
The commutative square
\[
\begin{tikzcd}
n\Cat\ar[r] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r] & (n \shortminus 1)\Mag
\end{tikzcd}
\]
induces a canonical functor
\[
V : n\Cat \to n\PCat,
\]
which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.
Moreover, the canonical commutative diagram
\[
\begin{tikzcd}
n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\
n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l]
\end{tikzcd}
\]
induces a canonical functor
\[
W : n\PCat \to n\CellExt.
\]
For an $n$-precategory $C$, $W(C)$ is simply the cellular extension $(C_n,\tau(C),s,t)$.
Finally, we define the functor
\[
U := W \circ V : n\Cat \to n\CellExt.
\]
We will now explicitely construct a left adjoint of $U$. In order to do that, we will construct left adjoints of $W$ and $V$.
\end{paragr}
\begin{paragr}
Let $(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has:
\begin{itemize}
\item[-] a symbol $\hat{x}$ for each $x \in \Sigma$,
\item[-] a symbol $\hat{\comp_k}$ for each $k<n$,
\end{itemize}
\end{paragr}
\section{$\oo$-categories}
\begin{paragr}
For any $n>0$, there is an obvious ``truncation'' functor
\[
\nGrph \to (n\shortminus 1)\Grph
\]
that simply forgets the $n$-cells. We define the category $\oo\Grph$ of $\omega$-graphs as the limit of the diagram
\[
\cdots \to \nGrph \to (n\shortminus 1)\Grph \to \cdots \to 1\Grph \to 0\Grph.
\]
\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}
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}\label{paragr:defoocat}
Once again, for any $n>0$ there is a canonical ``truncation'' functor
\[
......@@ -311,97 +427,3 @@
\begin{proof}
\todo{...}
\end{proof}
\section{Generating cells}
\begin{paragr}
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\[
\begin{tikzcd}
n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Grph \ar[r] & (n \shortminus 1)\Grph.
\end{tikzcd}
\]
More concretely, an $n$-cellular extension can be encoded in the data of a quadruple $(\Sigma,C,s,t)$ where $\Sigma$ is a set, $C$ is a $(n\shortminus 1)$-category, $s$ and $t$ are maps
\[
s,t : \Sigma \to C_n
\]
such that
\[
\begin{tikzcd}
C_{n-1} & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] C_n & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] \Sigma
\end{tikzcd}
\]
satisfy the globular identities.
Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with extra $n$-cells that make it a $n$-graph.
A morphism of $n$-cellular extensions from $(\Sigma,C,s,t)$ to $(\Sigma',C',s',t')$ consists of a pair $(\varphi,f)$ where $f : C \to C'$ is a morphism of $(n\shortminus 1)\Cat$ and $\varphi : \Sigma \to \Sigma'$ is a map such that the squares
\[
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\text{ and }
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"t"] & \Sigma' \ar[d,"t'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\]
commute.
%From now on, we will denote such an object of $n\GCat$ by
%\[
%\begin{tikzcd}
% \Sigma \ar[r,shift left,"s"] \ar[r,shift right,"t"']& C
% \end{tikzcd}
%\]
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\[
\begin{tikzcd}
n\PCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Mag \ar[r] & (n \shortminus 1)\Mag.
\end{tikzcd}
\]
More concretely, objects of $n\PCat$ can be seen as $n$-magmas such that if we forget their $n$-cells then they satisfy the axioms of $(n\shortminus 1)$-category. The left vertical arrow of the previous square is easily seen to be full, and we will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$.
The commutative square
\[
\begin{tikzcd}
n\Cat\ar[r] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r] & (n \shortminus 1)\Mag
\end{tikzcd}
\]
induces a canonical functor
\[
V : n\Cat \to n\PCat,
\]
which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.
Moreover, the canonical commutative diagram
\[
\begin{tikzcd}
n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\
n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l]
\end{tikzcd}
\]
induces a canonical functor
\[
W : n\PCat \to n\CellExt.
\]
For an $n$-precategory $C$, $W(C)$ is simply the cellular extension $(C_n,\tau(C),s,t)$.
Finally, we define the functor
\[
U := W \circ V : n\Cat \to n\CellExt.
\]
We will now explicitely construct a left adjoint of $U$. In order to do that, we will construct left adjoints of $W$ and $V$.
\end{paragr}
\begin{paragr}
Let $(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has:
\begin{itemize}
\item[-] a symbol $\hat{x}$ for each $x \in \Sigma$,
\item[-] a symbol $\hat{\comp_k}$ for each $k<n$,
\end{itemize}
\end{paragr}
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