Commit 9af933dd authored by Leonard Guetta's avatar Leonard Guetta
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I have to stop due to teaching duties. First thing to do during the next...

I have to stop due to teaching duties. First thing to do during the next session : improve paragraph 1.2.2 in the style of 1.2.1. Then I have to finish this section. After that, it seems logical to me to write a section where I define the notion of 'weakly generating set of the k-cells' and 'basis of the k-cells' of a n-category and then a section where I explain how to compute colimits in nCat. Maybe the order should be 'section on basis' then 'section on generating cells' then 'section on colimits'.
parent e0cb1427
......@@ -41,6 +41,17 @@
%%%%%% MACROS %%%%%%%
%Tikz customs
%% the folowing is used to draw adjoint pairs
\tikzset{%
symbol/.style={%
draw=none,
every to/.append style={%
edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
}
}
% oo-categories
\newcommand{\oo}{\omega}
\newcommand{\Cat}{\mathbf{Cat}}
......
......@@ -220,25 +220,58 @@
We also define an $n$-category $\kappa(C)$ with
\begin{itemize}
\item[-]\tau(\kappa(C))=C,
\item[-]
\end{itemize}
\item[-]$\tau(\kappa(C))=C$,
\item[-]$\kappa(C)_n=\left\{(a,b) \in C_{n-1}\times C_{n-1} \vert a \text{ and } b \text{ are parallel}\right\}$,
\item[-]for $(a,b) \in \kappa(C)_n$,
\[
t((a,b))=a \text{ and } s((a,b))=b,
\]
\item[-] for $(a,b)$ and $(b,c) \in \kappa(C)_n$,
\[
(a,b)\comp_{n-1}(b,c) = (a,c),
\]
\item[-] for $(a,b)$ and $(a',b') \in \kappa(C)_n$ that are $k$-composable with $k<n-1$,
\[
(a,b)\comp_k(a',b') = (a\comp_k a',b\comp_k b').
\]
(The right hand side makes sense since $a$ and $b$ are parallel and, $a'$ and $b'$ are parallel.)
\end{itemize}
It is straightforward to check that $\kappa(C)$ is indeed an $n$-category and the correspondance $C \mapsto \kappa(C)$ can canonically be made into a functor:
\[
\kappa : (n\shortminus 1)\Cat \to n\Cat.
\]
\end{paragr}
\begin{lemma}
We have a sequence of adjunctions
\[
\iota \dashv \tau \dashv \kappa.
\iota \dashv \tau \dashv \kappa.
\]
Moreover, the unit of the adjunction $\iota \dashv \tau$ is the equality:
\[
\mathrm{id}_{(n\shortminus 1)\Cat}=\tau \circ \iota,
\]
\end{lemma}
and the co-unit of the right adjunction $\tau\dashv \kappa$ is the equality:
\[
\tau \circ \kappa = \mathrm{id}_{(n\shortminus 1)\Cat}.
\]
\end{lemma}
\begin{proof}
\todo{À écrire.}
\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}
\section{Generating cells}
\begin{paragr}
Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
\[
\begin{equation}\label{squarecellext}
\begin{tikzcd}
n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Grph \ar[r] & (n \shortminus 1)\Grph.
n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph,
\end{tikzcd}
\]
\end{equation}
where the right vertical arrow is the obvious forgetful functor.
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
......@@ -253,6 +286,13 @@ n\Grph \ar[r] & (n \shortminus 1)\Grph.
Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with extra $n$-cells that make it a $n$-graph.
We will sometimes write
\[
\begin{tikzcd}
C &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] \Sigma
\end{tikzcd}
\]
to denote an $n$-cellular extension $(\Sigma,C,s,t)$. \remtt{Est-ce que je garde cette notation ?}
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
\[
......@@ -267,19 +307,22 @@ n\Grph \ar[r] & (n \shortminus 1)\Grph.
\end{tikzcd}
\]
commute.
%From now on, we will denote such an object of $n\GCat$ by
%\[
%\begin{tikzcd}
% \Sigma \ar[r,shift left,"s"] \ar[r,shift right,"t"']& C
% \end{tikzcd}
%\]
Once again, we will use the notation $\tau$ for the functor
\[
\begin{aligned}
\tau : n\CellExt &\to (n\shortminus 1)\Cat\\
(\Sigma,C,s,t) &\mapsto C
\end{aligned}
\]
which is simply the top horizontal arrow of square \eqref{squarecellext}.
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\[
\begin{tikzcd}
n\PCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Mag \ar[r] & (n \shortminus 1)\Mag.
n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag.
\end{tikzcd}
\]
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 $n\Mag$.
......
......@@ -166,3 +166,8 @@
\]
\end{enumerate}
\end{paragr}
\[
\begin{tikzcd}[column sep=huge]
n\Cat \ar[r,"\tau",""{name=B,above},""{name=C,below}] & (n\shortminus 1)\Cat \ar[l,bend right,"\iota"',""{name=A, below}] \ar[l,bend left,"\kappa",""{name=D, above}] \ar[from=A,to=B,symbol=\dashv]\ar[from=C,to=D,symbol=\dashv].
\end{tikzcd}
\]
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