### Il faut que j'avance plus vite!

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 \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 $k0$, 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
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