### ça risque d'être long

parent 388a2c62
 \chapter{$\oo$-Category theory} \chapter{Yoga of $\oo$-Categories} \section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories} \begin{paragr}\label{pargr:defngraph} ... ... @@ -316,8 +316,8 @@ Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product $\begin{tikzcd} n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& n\Grph \ar[d] \\ (n \shortminus 1)\Cat \ar[r] & (n \shortminus 1)\Grph. n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\ n\Grph \ar[r] & (n \shortminus 1)\Grph. \end{tikzcd}$ 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 ... ... @@ -332,7 +332,7 @@ \] satisfy the globular identities. Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with some extra $n$-cells that makes it a $n$-graph. 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 ... ... @@ -363,7 +363,7 @@ 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, 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 $n\Mag$. 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 ... ... @@ -377,13 +377,31 @@ $V : n\Cat \to n\PCat,$ which is also easily seen to be full. which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$. Finally, the canonical commutative diagram Moreover, the canonical commutative diagram $\begin{tikzcd} sd n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\ n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l] \end{tikzcd}$ \end{paragr} 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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