Commit a29f27e9 authored by Leonard Guetta's avatar Leonard Guetta
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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 $k<n$,
\end{itemize}
\end{paragr}
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