Commit 63d66e18 authored by Leonard Guetta's avatar Leonard Guetta
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parent 9af933dd
......@@ -260,7 +260,15 @@
\end{proof}
\begin{remark}
The ``truncation'' functors $\tau : n\Mag \to (n\shortminus 1)\Mag$ and $\tau : n\Grph \to (n\shortminus 1)\Grph$ also have left and right adjoints but we won't need them in the sequel.
\end{remark}
\end{remark}
\section{Basis for $n$-categories}
\begin{paragr}
Let $C$ be an $n$-category and $k\in \mathbb{N}$ with $k<n$. A \emph{$k$-prebasis} of $C$ is a subset
\[
\Sigma \subseteq C_k
\]
such that $C_k$ is the smallest subset
\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
......@@ -319,23 +327,26 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph,
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\[
\begin{equation}\label{squareprecat}
\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,"\tau"] & (n \shortminus 1)\Mag.
\end{tikzcd}
\end{equation}
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 top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation
\[
\tau : n\PCat \to (n\shortminus 1)\Cat.
\]
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
\[
\begin{tikzcd}
n\Cat\ar[r] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r] & (n \shortminus 1)\Mag
n\Cat\ar[r,"\tau"] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag,
\end{tikzcd}
\]
induces a canonical functor
where the vertical arrows are the obvious forgetful functors, induces a canonical functor
\[
V : n\Cat \to n\PCat,
\]
......@@ -352,19 +363,32 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph,
\[
W : n\PCat \to n\CellExt.
\]
For an $n$-precategory $C$, $W(C)$ is simply the cellular extension $(C_n,\tau(C),s,t)$.
For an $n$-precategory $C$, $W(C)$ is simply the cellular extension
\[
\begin{tikzcd}
\tau(C) &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] C_n.
\end{tikzcd}
\]
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 $V$.
The relation between $n\Cat$, $n\PCat$, $n\CellExt$ and $(n\shortminus 1)\Cat$ is summed up in the following commutative diagram:
\[
\begin{tikzcd}
n\Cat \ar[rr,bend left,"U"]\ar[r,"V"] \ar[rrd,"\tau"'] & n\PCat \ar[r,"W"]\ar[rd,"\tau"] & n\CellExt \ar[d,"\tau"] \\
&&(n\shortminus 1)\Cat.
\end{tikzcd}
\]
We will now explicitely construct a left adjoint of $U$. In order to do that, we will successively 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 $k<n$,
\item[-] a symbol
\end{itemize}
\end{paragr}
\section{$\oo$-categories}
......
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