### 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 $kk \in \mathbb{N}$, 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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