Commit 298d099e authored by Leonard Guetta's avatar Leonard Guetta
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Je prepare le terrain pour toute la partie sur le calcul des colimites dans...

Je prepare le terrain pour toute la partie sur le calcul des colimites dans ooCat. Il va peut etre falloir introduire plus de notations
parent 70c44319
......@@ -52,6 +52,8 @@
\newcommand{\ooMag}{\mathbf{\oo Mag}}
\newcommand{\nMag}{n \mathbf{Mag}}
\newcommand{\Mag}{\mathbf{Mag}}
\newcommand{\PCat}{\mathbf{PCat}}
\newcommand{\GCat}{\mathbf{GCat}}
% compositions and units
\def\1^#1_#2{1^{(#1)}_{#2}}
......
......@@ -214,5 +214,77 @@
that simply forgets the $n$-cells. We define the category $\oo\Cat$ of $\oo$-categories as the limit of the diagram
\[
\cdots \to n\Cat \to (n\shortminus 1)\Cat \to \cdots \to 1\Cat \to 0\Cat.
\]
By definition, for every $n \in \mathbb{N}$, there is a canonical forgetful functor
\[
n\Cat \to n\Mag,
\]
which is full. When $n>0$, the canonical square
\[
\begin{tikzcd}
n\Cat\ar[r] \ar[d] & n\Mag \ar[d]\\
(n \shortminus 1)\Cat \ar[r] & (n \shortminus 1)\Mag
\end{tikzcd}
\]
is commutative. By universal property, this yields a canonical forgetful functor
\[
\oo\Cat \to \oo\Mag,
\]
which is easily seen to be full.
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\GCat$ as the following fibred product
\[
\begin{tikzcd}
n\GCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& n\Grph \ar[d] \\
(n \shortminus 1)\Cat \ar[r] & (n \shortminus 1)\Grph.
\end{tikzcd}
\]
More concretely, an object of $n\GCat$ consists of 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
\[
s,t : \Sigma \to C_n
\]
such that
\[
\begin{tikzcd}
C_{n-1} & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] C_n & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] \Sigma
\end{tikzcd}
\]
satisfy the globular identities.
A morphism of $n\GCat$ 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
\[
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\]
%From now on, we will denote such an object of $n\GCat$ by
%\[
%\begin{tikzcd}
% \Sigma \ar[r,shift left,"s"] \ar[r,shift right,"t"']& C
% \end{tikzcd}
%\]
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\[
\begin{tikzcd}
n\PCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
n\Mag \ar[r] & (n \shortminus 1)\Mag.
\end{tikzcd}
\]
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 commutative square
\[
\begin{tikzcd}
n\Cat\ar[r] \ar[d] & n\Mag \ar[d]\\
(n \shortminus 1)\Cat \ar[r] & (n \shortminus 1)\Mag
\end{tikzcd}
\]
induces a canonical functor
\[
U : n\Cat \to n\PCat,
\]
which is easily seen to be full.
\end{paragr}
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