Commit e0cb1427 authored by Leonard Guetta's avatar Leonard Guetta
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parent 0c97b4a7
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\1^{k}_x := x.
A cell is \emph{degenerate} if it is a unit on a strictly lower dimensional cell.
A cell is \emph{degenerate} or \emph{trivial} if it is a unit on a strictly lower dimensional cell.
Let $C$ and $C'$ be $n$-magmas. A \emph{morphism of $n$-magmas} $f : C \to C'$ is a morphism of $n$-graphs that is compatible with compositions and units. This means:
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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.
Let $n>0$. As in the case of $n$-graphs and $n$-magmas, there is an obvious ``truncation'' functor
\tau : n\Cat \to (n\shortminus 1)\Cat
that simply forgets the $n$-cells and the square
n\Cat \ar[r,"\tau"]\ar[d] & (n\shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r,"\tau"] & (n\shortminus 1)\Mag
where the vertical arrows are the obvious forgetul functors, is commutative.
Now let $C$ be an $(n\shortminus 1)$-category. We define an $n$-category $\iota(C)$ with
\item[-] $\tau(\iota(C))=C$,
\item[-] $\iota(C)_{n}=C_{n-1}$,
\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 $k<n$, composition map $\iota(C)_n\underset{\iota(C)_k}{\times}\iota(C)_n \simeq \iota(C)_n \to \iota(C)_n$ as the identity.
It is immediate to see that $\iota(C)$ is indeed an $n$-category and the correspondance $ C \mapsto \iota(C)$ can canonically be made into a functor:
\iota : (n\shortminus1)\Cat \to n\Cat.
% andy definition, for any $(n \shortminus 1)$-category, we have
% \[
% \tau(\iota(C))=C.
% \]
We also define an $n$-category $\kappa(C)$ with
We have a sequence of adjunctions
\iota \dashv \tau \dashv \kappa.
\section{Generating cells}
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
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