Commit f2a17372 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Slowly but surely

parent d7b23e1d
@article{bidon,
title={citation bidon}
}
\ No newline at end of file
......@@ -9,7 +9,7 @@
\maketitle
\include{omegacat}
\cite{bidon}
\include{homtheo}
\bibliographystyle{alpha}
\bibliography{main}
\bibliography{memoire}
\end{document}
......@@ -13,6 +13,9 @@
\usepackage{tikz-cd}
% List
\usepackage{enumitem}
% Theorems
\theoremstyle{plain}
......@@ -30,7 +33,7 @@
\newtheorem*{notation}{Notation}
\newtheorem{paragr}[theorem]{}
\newtheorem{example}[theorem]{Example}
\newtheorem{impexample}[theorem]{Important example}
\theoremstyle{plain}
\newtheorem*{theorem*}{Theorem} % Unnumbered theorem
\newtheorem{theoremintro}{Theorem} % Theorem environment for the introduction. Numbering not following the sections
......@@ -65,6 +68,10 @@
\newcommand{\T}{\ensuremath{\mathcal{T}}} %Idem
\newcommand{\G}{\ensuremath{\mathcal{G}}} %Idem
%Small categories
\newcommand{\sD}{\ensuremath{\mathbb{D}}}
% oo-categories
\newcommand{\oo}{\omega}
\newcommand{\Cat}{\mathbf{Cat}}
......@@ -79,6 +86,8 @@
\newcommand{\PCat}{\mathbf{PCat}}
\newcommand{\CellExt}{\mathbf{CellExt}}
\newcommand{\CCat}{\underline{\mathbf{Cat}}} %2-category of small categories
\newcommand{\CCAT}{\underline{\mathbf{CAT}}} %2-category of big categories
% compositions and units
\def\1^#1_#2{1^{(#1)}_{#2}} % for iterated units
......@@ -101,6 +110,19 @@
% Maths
\DeclareMathSymbol{\shortminus}{\mathbin}{AMSa}{"39} %For short minus signs
\newcommand{\Hom}{\ensuremath{\mathrm{Hom}}}
\newcommand{\Ob}{\ensuremath{\mathrm{Ob}}}
\newcommand{\Psh}[1]{\ensuremath{\widehat{#1}}} %Presheaves categories
\newcommand{\id}{\ensuremath{\mathrm{id}}} %identity in a category
\def\colim{\mathop{\mathrm{colim}}} %colimits
\def\hocolim{\mathop{\mathrm{hocolim}}} %homotopy colimits
\newcommand{\wt}[1]{\ensuremath{\widetilde{#1}}} %A shortcut for \widetilde
\newcommand{\LL}{\ensuremath{\mathbb{L}}} % A mathbb L. Useful for left derived functor
\newcommand{\RR}{\ensuremath{\mathbb{R}}} % A mathbb R. Useful for right left derived functor
\newcommand{\Fib}{\ensuremath{\mathrm{Fib}}} % A mathrm "Fib". Used to denote fibrations in a model category.
\newcommand{\Cof}{\ensuremath{\mathrm{Cof}}} % A mathrm "Cof". Used to denote cofibrations in a model category.
\newcommand{\Ho}{\ensuremath{\mathrm{Ho}}} %Useful for the homotopy category
% commentaires
\newcommand\remtt[1]{\texttt{[#1]}}
......
......@@ -92,11 +92,11 @@
x &\mapsto 1_x
\end{aligned}
\]
for every $k \in \mathbb{N}$ with $k\leq n$,
for every $k \in \mathbb{N}$ with $k< n$,
\end{itemize}
subject to the following axioms:
\begin{itemize}
\item[-] for all $k,l \in \mathbb{N}$ with $k<l\leq n$ and every $k$-composable $l$-cells $x$ and $y$,
\begin{enumerate}[label=(\alph*)]
\item For all $k,l \in \mathbb{N}$ with $k<l\leq n$ and every $k$-composable $l$-cells $x$ and $y$,
\[
s(x\underset{k}{\ast} y) =
\begin{cases}
......@@ -112,11 +112,11 @@
t(x)\underset{k}{\ast} t(y) &\text{ otherwise.}
\end{cases}
\]
\item[-]for every $k \in \mathbb{N}$ with $k\leq n$ and every $k$-cell,
\item For every $k \in \mathbb{N}$ with $k\leq n$ and every $k$-cell,
\[
s(1_x)=t(1_x)=x.
\]
\end{itemize}
\end{enumerate}
We will use the same letter to denote an $n$-magma and its underlying $n$-graph.
For two $k$-composable $l$-cells $x$ and $y$, we refer to $x\ast_ky$ as the \emph{$k$-composition} of $x$ and $y$.
......@@ -163,7 +163,7 @@
\begin{paragr}
Let $n \in \mathbb{N}$. An \emph{$n$-category} $C$ is an $n$-magma such that the following axioms are satisfied:
\begin{enumerate}
\begin{enumerate}[label=(\textbf{CAT}\alph*)]
\item for all $k,l \in \mathbb{N}$ with $k<l\leq n$, for all $k$-composable $l$-cells $x$ and $y$, we have
\[
1_{x\underset{k}{\ast}y}=1_{x}\underset{k}{\ast}1_{y},
......@@ -288,123 +288,23 @@
\end{definition}
\begin{definition}
\end{definition}
\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,"\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
\end{definition}
\section{Generating cells}
\begin{paragr}
A \emph{cellular extension} of an $n$-magma $M$ in the data of a quadruple $(\Sigma,M,s,t)$ where $\Sigma$ is a set, $M$ is a $n$-magma, $s$ and $t$ are maps
\[
s,t : \Sigma \to C_n
s,t : \Sigma \to M_n
\]
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.
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
\[
\begin{tikzcd}
C &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] \Sigma
M_{n-1} & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] M_n & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] \Sigma
\end{tikzcd}
\]
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
\[
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\text{ and }
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"t"] & \Sigma' \ar[d,"t'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\]
commute.
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}.
satisfy the globular identities. We will often use the abusive notation $(\Sigma,M)$ instead of $(\Sigma,M,s,t)$.
\end{paragr}
\begin{paragr}
Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
\begin{equation}\label{squareprecat}
\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,"\tau"] & (n \shortminus 1)\Mag.
\end{tikzcd}
\end{equation}
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 $n\Mag$. The top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation
\[
\tau : n\PCat \to (n\shortminus 1)\Cat.
\]
The commutative square
\[
\begin{tikzcd}
n\Cat\ar[r,"\tau"] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag,
\end{tikzcd}
\]
where the vertical arrows are the obvious forgetful functors, induces a canonical functor
\[
V : n\Cat \to n\PCat,
\]
which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.
Moreover, the canonical commutative diagram
\[
\begin{tikzcd}
n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\
n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l]
\end{tikzcd}
\]
induces a canonical functor
\[
W : n\PCat \to n\CellExt.
\]
For an $n$-precategory $C$, $W(C)$ is simply the cellular extension
\[
\begin{tikzcd}
\tau(C) &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] C_n.
\end{tikzcd}
\]
Finally, we define the functor
\[
U := W \circ V : n\Cat \to n\CellExt.
\]
The relation between $n\Cat$, $n\PCat$, $n\CellExt$ and $(n\shortminus 1)\Cat$ is summed up in the following commutative diagram:
\[
\begin{tikzcd}
n\Cat \ar[rr,bend left,"U"]\ar[r,"V"] \ar[rrd,"\tau"'] & n\PCat \ar[r,"W"]\ar[rd,"\tau"] & n\CellExt \ar[d,"\tau"] \\
&&(n\shortminus 1)\Cat.
\end{tikzcd}
\]
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 $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. Consider the alphabet that has:
Let $E=(\Sigma,M,s,t)$ be an cellular extension of an $n$-magma. Consider the alphabet that has:
\begin{itemize}
\item[-] a symbol $\hat{x}$ for each $x \in \Sigma$,
\item[-] a symbol $\fcomp_k$ for each $k<n$,
......@@ -415,39 +315,55 @@ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph,
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 $w$.
We now recursively define the set $\Sigma^{+} \subseteq \W[\Sigma]$ of \emph{well formed words} on this alphabet together with maps $s,t : \Sigma^{+} \to C_{n-1}$:
We now recursively define the set $\Sigma^{+} \subseteq \W[\Sigma]$ of \emph{well formed words} on this alphabet together with maps $s,t : \Sigma^{+} \to M_{n-1}$:
\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 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
We define $s_k , t_k: \Sigma^{+} \to M_k$ 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}
%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}
Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with
\begin{itemize}
\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 $k<n$,
Let $C$ be an $n$-category, $k \in \mathbb{N}$ with $0<k \leq n$ and $\Sigma \subseteq C_k$ a subset of the $k$-cells of $C$ with $k\leq n$. It defines an cellular extension $(\Sigma,\tau_{k-1}(C),\sigma,\tau)$ of $\tau_{k-1}(C)$, where $s$ and $t$ are simply the restriction of the source and target maps $C_k \to C_{k-1}$. The canonical inclusion
\[
v\comp_kw := (v \fcomp_k w).
\Sigma \hookrightarrow C_k
\]
\end{itemize}
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
is recursively extended to a map $\rho : \Sigma^{+} \to C_k$ in the following way:
\begin{enumerate}[label=-]
\item $\rho(\hat{x})=x$ for every $x \in \Sigma$,
\item $\rho((v\fcomp_kw))=\rho(v)\comp_k\rho(w)$.
\end{enumerate}
We call $\rho$ the \emph{evaluation map}. Intuitively, this maps ``evaluates'' the formal expressions constructed out of the cells in $\Sigma$ into actual $k$-cells of $\Sigma$.
\end{paragr}
\begin{definition}
Let $C$ be an $n$-category and $k\leq n$. A subset $\Sigma \subseteq C_k$ of $k$-cells \emph{generates by composition} if the evaluation map
\[
\rho : \Sigma^+ \to C_k
\]
is surjective.
\end{definition}
\begin{definition}
Let $M$ be an $n$-magma and $k \leq n$. A \emph{congruence} on the $k$-cells of $M$ is a binary relation $\R$ on $M_k$ such that
\begin{enumerate}[label=-]
\item $\R$ is an equivalence relation,
\item if $x\R y$ then $x$ and $y$ are parallel,
\item for $x, x', y, y' \in M_k$ such that $x$ and $x'$ are $l$-composable, $y$ and $y'$ are $l$-composable if $x\R y$ and $x'\R y'$, then $(x\ast_l y )\R (x'\ast_l y')$.
\end{enumerate}
\end{definition}
\begin{lemma}
Let $M$ be an $n$-magma and $(\R_i)_{i \in I}$ a family of congruence on the $k$-cells of $M$. If $I$ is non-empty then $\cap_{i \in I} \R_i$ is a congruence on the $k$-cells of $M$.
\end{lemma}
\
\section{$\oo$-categories}
\begin{paragr}
For any $n>0$, there is an obvious ``truncation'' functor
......
......@@ -192,3 +192,147 @@ and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\end{tikzcd}
\]
is an $n$-graph and
\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,"\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
\[
s,t : \Sigma \to C_n
\]
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.
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
\[
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\text{ and }
\begin{tikzcd}
\Sigma \ar[r,"\varphi"] \ar[d,"t"] & \Sigma' \ar[d,"t'"] \\
C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
\end{tikzcd}
\]
commute.
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{equation}\label{squareprecat}
\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,"\tau"] & (n \shortminus 1)\Mag.
\end{tikzcd}
\end{equation}
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 $n\Mag$. The top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation
\[
\tau : n\PCat \to (n\shortminus 1)\Cat.
\]
The commutative square
\[
\begin{tikzcd}
n\Cat\ar[r,"\tau"] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag,
\end{tikzcd}
\]
where the vertical arrows are the obvious forgetful functors, induces a canonical functor
\[
V : n\Cat \to n\PCat,
\]
which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.
Moreover, the canonical commutative diagram
\[
\begin{tikzcd}
n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\
n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l]
\end{tikzcd}
\]
induces a canonical functor
\[
W : n\PCat \to n\CellExt.
\]
For an $n$-precategory $C$, $W(C)$ is simply the cellular extension
\[
\begin{tikzcd}
\tau(C) &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] C_n.
\end{tikzcd}
\]
Finally, we define the functor
\[
U := W \circ V : n\Cat \to n\CellExt.
\]
The relation between $n\Cat$, $n\PCat$, $n\CellExt$ and $(n\shortminus 1)\Cat$ is summed up in the following commutative diagram:
\[
\begin{tikzcd}
n\Cat \ar[rr,bend left,"U"]\ar[r,"V"] \ar[rrd,"\tau"'] & n\PCat \ar[r,"W"]\ar[rd,"\tau"] & n\CellExt \ar[d,"\tau"] \\
&&(n\shortminus 1)\Cat.
\end{tikzcd}
\]
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 $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with
\begin{itemize}
\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}
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}
For $u : A \to B$ in $\CCat$, let
\[
u^* : \C(A) \to \C(B)
\]
be the functor induced by post-composition. For $\begin{tikzcd} \sD(B)\ar[r,bend left,"u^*",""{name=U,below}] \ar[r,bend right,"v^*"',""{name=D,above}] & \sD(A) \ar[from=U,to=D,Rightarrow,"\alpha^*"] \end{tikzcd}$
Note that we have a canonical isomorphism
\[
\C(e) \simeq \C
\]
and for any small category $A$, the functor
\[
p_A^* : \C(e) \to \C(A)
\]
is canonically isomorphic with the diagonal functor $\Delta : \C \to \C(A)$ that sends an object $X$ of $\C$ to the constant diagram $A \to \C$ with value $X$.
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