Commit d7b23e1d authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Fini pour ajd

parent 761cc007
......@@ -80,9 +80,12 @@
\newcommand{\CellExt}{\mathbf{CellExt}}
% compositions and units
\def\1^#1_#2{1^{(#1)}_{#2}}
\def\1^#1_#2{1^{(#1)}_{#2}} % for iterated units
\def\comp_#1{\underset{#1}{\ast}}
\def\comp_#1{\mathbin{\underset{#1}{\ast}}} % for composition
\def\fcomp_#1{\mathbin{\hat{\underset{#1}{\ast}}}} % formal composition
% useful stuff
......
......@@ -404,33 +404,49 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph,
We will now explicitely construct a left adjoint of $U$. In order to do that, we will successively construct left adjoints of $W$ and $V$.
\end{paragr}
\begin{paragr}
Let $(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has:
Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has:
\begin{itemize}
\item[-] a symbol $\hat{x}$ for each $x \in \Sigma$,
\item[-] a symbol $\hat{\comp_k}$ for each $k<n$,
\item[-] a symbol $\fcomp_k$ for each $k<n$,
\item[-] a symbol $\ii_x$ for each $x \in C_{n\shortminus 1}$,
\item[-] a symbol of opening parenthesis $($,
\item[-] a symbol of closing parenthesis $)$.
\end{itemize}
We denote by $\W[E]$ the set of words on this alphabet (i.e. finite sequence of symbols). If $w$ and $w'$ are elements of $\mathcal{W}[E]$, we write $ww'$ for their concatenation.
We denote by $\W[\Sigma]$ the set of words on this alphabet (i.e. finite sequence of symbols). If $w$ and $w'$ are elements of $\mathcal{W}[\Sigma]$, we write $ww'$ for their concatenation.
The \emph{length} of a word $w$, denoted by $\mathcal{L}(w)$, is the number of symbols that appear in $w$.
%The \emph{length} of a word $w$, denoted by $\mathcal{L}(w)$, is the number of symbols that appear in $w$.
We now recursively define the set $\Sigma^{+} \subseteq \W[\Sigma]$ of \emph{well formed words} on this alphabet together with maps $s,t : \Sigma^{+} \to C_{n-1}$:
\begin{itemize}
\item[-] for every $x \in \Sigma$, we have $(\hat{x}) \in \Sigma^{+}$ with
\[s((\hat{x}))=s(x) \text{ and }t((\hat{x}))=t(x),\]
\item[-] for every $x \in C_n$, we have $(\ii_{x}) \in \Sigma^{+}$ with
\[s((\ii_x))=t((\ii_x))=x,\]
\item[-] for all $v,w \in \Sigma^{+}$ such that $s(v)=t(w)$, we have $ (v \fcomp_n w) \in \Sigma^{+}$ with \[s((v \fcomp_n w))=s(w)\text{ and }t((v \fcomp_n w))=t(v),\]
\item[-] for all $v, w \in \Sigma^{+}$ and $0 \leq k < n\shortminus 1$, such that $s_k(s(v))=t_k(t(w))$, we have $(v \fcomp_k w) \in \Sigma^{+}$ with \[s((v \hat{\comp_k} w)) = s(v) \comp_k s(w)\] and \[t((v \hat{\comp_k} w))=t(v)\comp_k t(w).\]
\end{itemize}
We define $s_k , t_k: \Sigma^{+} \to C_k$ as iterated source and target (with $s_n=s$ and $t_n=t$ for consistency). We say that two well formed words $v$ and $w$ are \emph{parallel} if
\[s(v)=s(w) \text{ and }t(v)=t(w).\]
and we say that they are \emph{$k$-composable} for a $k< n$ if
\[s_k(v)=t_k(w).\]
Let $E'=(\Sigma',C',s',t')$ be another $n$-cellular extension and $(\varphi,f) : E \to E'$ a morphism of $n$-cellular extensions. We recursively define a map $f^+ : \Sigma^+ \to \Sigma^+$ with
\begin{itemize}
\item[-]
\end{itemize}
\end{paragr}
\begin{paragr}
We now recursively define the set $\Sigma^{+} \subseteq \W[\Sigma]$ of \emph{well formed words} (or \emph{terms}) on this alphabet together with maps $s,t : \Sigma^{+} \to C_{n-1}$ that satisfy the globular conditions:
\begin{paragr}
Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with
\begin{itemize}
\item[-] $(\hat{x}) \in \Sigma^{+}$ with
\[s((\hat{x}))=s(x) \text{ and }t((\hat{x}))=t(x)\]
for each $x \in \Sigma$,
\item[-] $(\ii_{x}) \in \Sigma^{+}$ with $s((\ii_x))=t((\ii_x))=x$ for each $x \in C_n$,
\item[-] $ (v \comp_n w) \in \Sigma^{+}$ with $s((v \comp_n w))=s(w)$ and $t((v \comp_n w))=t(v)$ for $v,w \in \Sigma^{+}$ such that $s(v)=t(w)$,
\item[-] $(v \hat{\comp_k} w) \in \Sigma^{+}$ with \[s((v \hat{\comp_k} w)) = s(v) \comp_k s(w)\] and \[t((v \hat{\comp_k} w))=t(v)\comp_k t(w)\] for $v, w \in \Sigma^{+}$ and $0 \leq k < n$, such that $s_k(s(v))=t_k(t(w))$.
\item[-] $\tau(W_!(E))=C$,
\item[-] $W_!(E)_n=\Sigma^{+}$,
\item[-] source and target maps $\Sigma^+ \to C_{n-1}$ as defined in the previous paragraph,
\item[-] for every $x \in C_{n-1}$,
\[1_x := (\ii_x)\]
\item[-] for every $v,w \in \Sigma^+$ that are $k$-composable for a $k<n$,
\[
v\comp_kw := (v \fcomp_k w).
\]
\end{itemize}
%As usual, if $w \in \Sigma^{+}$ we often write $w : x \to y$ to say that $s_n(w)=x$ and $t_n(w)=y$.
We define $s_k , t_k: \Sigma^{+} \to C_k$ as iterated source and target (with $s_n=s$ and $t_n=t$ for consistency). We say that two well formed words $v$ and $w$ are \emph{parallel} if
\[s(v)=s(w) \text{ and }t(v)=t(w)\]
and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\[s_k(v)=t_k(w).\]
It is straightforward to check that this defines an $n$-precategory. Let $(\varphi,f) : E \to E'$ be a morphism of $n$-cellular extensions. We define a morphism of $n$-precategories $W_!(\varphi,f) : W_!(E) \to W_!(E')$ with
\end{paragr}
\section{$\oo$-categories}
\begin{paragr}
......
......@@ -171,3 +171,24 @@
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}
\]
For every $k\in \mathbb{N}$ such that $k < n$, we define the set $\Sigma^{+}\underset{C_k}{\times}\Sigma^{+}$ as the following fibred product
\[
\begin{tikzcd}
\Sigma^{+}\underset{C_k}{\times}\Sigma^{+} \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &\Sigma^{+} \ar[d,"t_k"]\\
\Sigma^{+} \ar[r,"s_k"] & C_k.
\end{tikzcd}
\]
That is, elements of $\Sigma^{+}\underset{C_k}{\times}\Sigma^{+}$ are pairs $(x,y)$ of well formed words such that $s_k(x)=t_k(y)$. We say that two well formed words $x$ and $y$ are \emph{$k$-composable} if the pair $(x,y)$ belongs to $\Sigma^{+}\times_{C_k}\Sigma^{+}$.
\end{paragr}
We define $s_k , t_k: \Sigma^{+} \to C_k$ as iterated source and target (with $s_n=s$ and $t_n=t$ for consistency). We say that two well formed words $v$ and $w$ are \emph{parallel} if
\[s(v)=s(w) \text{ and }t(v)=t(w).\]
and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\[s_k(v)=t_k(w).\]
It is straightforward to check that
\[
\begin{tikzcd}
C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] \cdots & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_{n \shortminus 1} & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] \Sigma^{+}
\end{tikzcd}
\]
is an $n$-graph and
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