Commit 866dd53e authored by Leonard Guetta's avatar Leonard Guetta
Browse files

changing computers

parent aca3dd48
......@@ -64,55 +64,44 @@
\end{tikzcd},
\]
\[
\Or_3=\begin{tikzcd}[column sep=huge,row sep=huge]
\langle 0 \rangle \ar[r,"\langle 01 \rangle"description] \ar[d,"\langle 13 \rangle"description] & \langle 1 \rangle \ar[d,"\langle 02 \rangle"description]\\
\langle 3 \rangle & \langle 3 \rangle \ar[l,"\langle 23 \rangle" description]
\ar[from=1-1,to=2-2,"\langle 02 \rangle" description,""{name=A,above}]
\ar[from=1-2,to=A,Rightarrow,"\langle 013 \rangle"description]
\end{tikzcd}
\overset{\langle 0123 \rangle}{\Rrightarrow}
\begin{tikzcd}[column sep=huge,row sep=huge]
\langle 0 \rangle \ar[r,"\langle 01 \rangle"description] \ar[d,"\langle 13 \rangle"description] & \langle 1 \rangle \ar[d,"\langle 02 \rangle"description]\\
\langle 3 \rangle & \langle 3 \rangle \ar[l,"\langle 23 \rangle" description]
\ar[from=1-2,to=2-1,"\langle 12 \rangle" description]
\end{tikzcd}
\]
\[
\Or_3=
\begin{tikzcd}
& \langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=B,above}] \ar[rr,"\langle 02 \rangle" description,""{name=A,above}]& & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\
& \langle 3 \rangle &
\ar[from=A,to=1-2,Rightarrow,"\langle 012 \rangle"]
\ar[from=B,to=2-3,Rightarrow,"\langle 023 \rangle" description, shorten <= 1em, shorten >= 1em]
\ar[from=A,to=1-2,Rightarrow,"\langle 012 \rangle", shorten <= 0.25em, shorten >= 0.25em]
\ar[from=B,to=2-3,Rightarrow,"\langle 023 \rangle"', near start, shorten <= 1.1em, shorten >= 1.5em]
\end{tikzcd}
\overset{\langle 0123 \rangle}{\Rrightarrow}
\begin{tikzcd}
& \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description] & \\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"'] & & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\
& \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description,""{name=B,right}] & \\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=A,above}] & & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\
& \langle 3 \rangle &
\ar[from=A,to=1-2,Rightarrow,"\langle 013 \rangle", near start, shorten <= 1em, shorten >= 1.5em]
\ar[from=B,to=2-3,Rightarrow,"\langle 123 \rangle", shorten <= 0.75em, shorten >=0.75em]
\end{tikzcd}
\]
\end{paragr}
\begin{paragr}\label{paragr:nerve}
For every $\omega$-category $X$, the \emph{nerve of $X$} is the simplicial set $N_{\omega}(X)$ defined as
For every $\omega$-category $C$, the \emph{nerve of $C$} is the simplicial set $N_{\omega}(C)$ defined as
\[
\begin{aligned}
N_{\omega}(X) : \Delta^{op} &\to \Set\\
[n] &\mapsto \Hom_{\omega\Cat}(\Or_n,X).
N_{\omega}(C) : \Delta^{op} &\to \Set\\
[n] &\mapsto \Hom_{\omega\Cat}(\Or_n,C).
\end{aligned}
\]
By post-composition, this yields a functor
\[
\begin{aligned}
N_{\omega} : \omega\Cat &\to \Psh{\Delta} \\
X &\mapsto N_{\omega}(X),
C &\mapsto N_{\omega}(C),
\end{aligned}
\]
and we define for every $n \in \mathbb{N}$, a nerve functor for $n$-categories $N_n : n\Cat \to \Psh{\Delta}$ as the following composite
which we refer to as the \emph{nerve functor for $\oo$-categories}. Furthermore, for every $n \in \mathbb{N}$, we also define a nerve functor for $n$-categories as the restriction of $N_{\oo}$ to $n\Cat$ (seen as a full subcategory of $\oo\Cat$)
\[
n\Cat \hookrightarrow \oo\Cat \overset{N_{\oo}}{\longrightarrow} \Psh{\Delta}.
N_n := N_{\oo}\vert_{n\Cat} : n\Cat \to \Psh{\Delta}.
\]
By the usual Kan extension technique, we obtain for each $n \in \nbar$, a functor \[c_n : \Psh{\Delta} \to n\Cat,\] left adjoint of $N_n$.
By the usual Kan extension technique, we obtain for each $n \in \nbar$ a functor \[c_n : \Psh{\Delta} \to n\Cat,\] left adjoint of $N_n$.
\end{paragr}
\begin{lemma}
Let $X$ be a simplicial set. The $\oo$-category $c_{\oo}(X)$ is free and the set of generating $k$-cells of $c_{\oo}(X)$ is canonically isomorphic the to set of non-degenerate $k$-simplices of $X$.
......
......@@ -679,3 +679,18 @@ In this vast generalization of category theory, the objects of study have object
\end{itemize}
\fi
%%%%% 3rd oriental
\begin{tikzcd}[column sep=huge,row sep=huge]
\langle 0 \rangle \ar[r,"\langle 01 \rangle"description] \ar[d,"\langle 13 \rangle"description] & \langle 1 \rangle \ar[d,"\langle 02 \rangle"description]\\
\langle 3 \rangle & \langle 3 \rangle \ar[l,"\langle 23 \rangle" description]
\ar[from=1-1,to=2-2,"\langle 02 \rangle" description,""{name=A,above}]
\ar[from=1-2,to=A,Rightarrow,"\langle 013 \rangle"description]
\end{tikzcd}
\overset{\langle 0123 \rangle}{\Rrightarrow}
\begin{tikzcd}[column sep=huge,row sep=huge]
\langle 0 \rangle \ar[r,"\langle 01 \rangle"description] \ar[d,"\langle 13 \rangle"description] & \langle 1 \rangle \ar[d,"\langle 02 \rangle"description]\\
\langle 3 \rangle & \langle 3 \rangle \ar[l,"\langle 23 \rangle" description]
\ar[from=1-2,to=2-1,"\langle 12 \rangle" description]
\end{tikzcd}
\]
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