Commit 388a2c62 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

sloooowly but surely (?)

parent 9a578f87
......@@ -53,7 +53,7 @@
\newcommand{\nMag}{n \mathbf{Mag}}
\newcommand{\Mag}{\mathbf{Mag}}
\newcommand{\PCat}{\mathbf{PCat}}
\newcommand{\GCat}{\mathbf{GCat}}
\newcommand{\CellExt}{\mathbf{CellExt}}
% compositions and units
\def\1^#1_#2{1^{(#1)}_{#2}}
......@@ -62,6 +62,10 @@
% ad hoc
\newcommand{\nbar}{\mathbb{N}\cup\{ \oo \}}
% squelette
\newcommand{\sk}{\mathrm{sk}}
% Maths
\DeclareMathSymbol{\shortminus}{\mathbin}{AMSa}{"39} %For short minus signs
......
......@@ -232,27 +232,95 @@
\]
which is easily seen to be full.
\end{paragr}
\remtt{Le lemme suivant est un genre d'Eilenberg-Zilber pour les oo-catégories.}
\begin{lemma}\label{EZoocat}
Let $C$ be an $\oo$-category and $n \in \mathbb{N}$. For any $n$-cell $x$ of $C$, there exist a unique non-degenerate $k$-cell $x'$ with $k\leq n$ such that
\[
\1^n_{x'}=x.
\]
\end{lemma}
\begin{proof}
\todo{Évident. À écrire ?}
\end{proof}
\begin{paragr}
For any $n>0$, let us denote $\tau$ the canonical truncation functor
\[
\tau : n\Cat \to (n \shortminus 1)\Cat
\]
from Paragraph \ref{paragr:defoocat} and
from Paragraph \ref{paragr:defoocat}.
Let $C$ be a $(n\shortminus 1)$-category. We define an $n$-magma $\iota(C)$ with
\begin{itemize}
\item[-] $\tau(\iota(C))=C$,
\item[-] $C'_{n}=C_{n-1}$,
\item[-] source and targets maps $\iota(C)_{n} \to \iota(C)_{n-1}$ as the identity,
\item[-] unit map $\iota(C)_{n-1} \to \iota(C)_n$ as the identity,
\item[-] for every $k<n$, composition map $\iota(C)_n\underset{\iota(C)_k}{\times}\iota(C)_n \simeq \iota(C)_n \to \iota(C)_n$ as the identity.
\end{itemize}
It is immediate to see that $\iota(C)$ is in fact an $n$-category and the correspondance $ C \mapsto \iota(C)$ can canonically made into a functor:
\[
\iota : (n\shortminus1)\Cat \to n\Cat.
\]
% By definition, we have
% \[
% \tau \circ \iota = \mathrm{id}_{(n\shortminus 1)\Cat}.
% \]
More generally, for all $n>k \in \mathbb{N}$, we define $\iota^n_k : k\Cat \to n\Cat$ as the composition
\[
\iota^n_k = \underbrace{\iota \circ \cdots \circ\iota}_{n-k \text{ times}}.
\]
For any $n \in \mathbb{N}$, the commutative diagram
\[
\begin{tikzcd}
\cdots \ar[r]&(n+3)\Cat \ar[r,"\tau"] & (n+2)\Cat \ar[r,"\tau"] &(n+1)\Cat\\
&&&\\
n\Cat \ar[uur,"\iota^{n+3}_n"] \ar[uurr,"\iota^{n+2}_n"] \ar[uurrr,"\iota^{n+1}_n"'] &&&
\end{tikzcd}
\]
induces by universal property a functor
\[
\iota^n : n\Cat \to \oo\Cat.
\]
Let us denote by
\[
\tau_{\leq n} : \oo\Cat \to n\Cat
\]
the canonical arrow of the limiting cone from the same paragraph.
Let $C$ be a $(n\shortminus 1)$-category. We define a category $C'$
the canonical arrow of the limiting cone.
\todo{Montrer qu'on a une adjonction $\iota_n \dashv \tau_{\leq n}$}
Let $C$ be an $\oo$-category. We define the \emph{$n$-skeleton of $C$} as the $\oo$-category
\[
\sk_n(C):= \iota^n(\tau_{\leq n} (C)).
\]
There is a canonical diagram in $\oo\Cat$ \todo{Comment on le définit ?}
\[
\begin{tikzcd}[column sep=small]
\sk_0(C) \ar[r] & \sk_1(C) \ar[r] & \cdots \ar[r] & \sk_n(C) \ar[r] &\sk_{n+1}(C) \ar[r] & \cdots
\end{tikzcd}
\]
that we call the \emph{canonical filtration of $C$} and the sequence of arrows
\[
(\eta_n : \sk_n(C) \to C)_{n \in \mathbb{N}}
\]
where $\eta_n$ is the unit of the adjunction $\iota^n \dashv \tau_n$, defines a cocone on the previous diagram.
\end{paragr}
\begin{lemma}
Every $\oo$-category is the colimit of its canonical filtration.
\end{lemma}
\begin{proof}
\todo{...}
\end{proof}
\section{Generating cells}
\begin{paragr}
Let $n>0$, we define the category $n\GCat$ as the following fibred product
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\[
\begin{tikzcd}
n\GCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& n\Grph \ar[d] \\
n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& n\Grph \ar[d] \\
(n \shortminus 1)\Cat \ar[r] & (n \shortminus 1)\Grph.
\end{tikzcd}
\]
More concretely, an object of $n\GCat$ consists of 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
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
\[
s,t : \Sigma \to C_n
\]
......@@ -263,13 +331,23 @@
\end{tikzcd}
\]
satisfy the globular identities.
A morphism of $n\GCat$ 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
Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with some extra $n$-cells that makes 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
\[
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\]
\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
%\[
%\begin{tikzcd}
......@@ -285,18 +363,27 @@
n\Mag \ar[r] & (n \shortminus 1)\Mag.
\end{tikzcd}
\]
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.
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, hence a morphism of $n$-precategories is just a morphism of underlying $n$-magmas. We will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$.
The commutative square
\[
\begin{tikzcd}
n\Cat\ar[r] \ar[d] & n\Mag \ar[d]\\
(n \shortminus 1)\Cat \ar[r] & (n \shortminus 1)\Mag
n\Cat\ar[r] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r] & (n \shortminus 1)\Mag
\end{tikzcd}
\]
induces a canonical functor
\[
U : n\Cat \to n\PCat,
V : n\Cat \to n\PCat,
\]
which is also easily seen to be full.
Finally, the canonical commutative diagram
\[
\begin{tikzcd}
sd
\end{tikzcd}
\]
which is easily seen to be full.
\end{paragr}
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