Commit 9af933dd by Leonard Guetta

### 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 $k0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product $\begin{equation}\label{squarecellext} \begin{tikzcd} n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\ n\Grph \ar[r] & (n \shortminus 1)\Grph. n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph, \end{tikzcd}$ 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} $\begin{tikzcd}[column sep=huge] n\Cat \ar[r,"\tau",""{name=B,above},""{name=C,below}] & (n\shortminus 1)\Cat \ar[l,bend right,"\iota"',""{name=A, below}] \ar[l,bend left,"\kappa",""{name=D, above}] \ar[from=A,to=B,symbol=\dashv]\ar[from=C,to=D,symbol=\dashv]. \end{tikzcd}$
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment