Commit 498aacaf authored by Leonard Guetta's avatar Leonard Guetta
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security commit

parent 56004c82
......@@ -123,6 +123,8 @@
% useful stuff
\newcommand{\ii}{\mathbf{i}} % a boldfont i
\newcommand{\cc}{\mathbf{c}} % a boldfont c
\newcommand{\bs}[1]{\ensuremath{\boldsymbol{#1}}} % a shortcut for \boldsymbol
\newcommand{\nbd}{\nobreakdash} % A shortcut for \nobreakdash
......
......@@ -147,7 +147,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\end{itemize}
we have
\[
((x \comp_k x')\comp_l (y \comp_k y'))=((x \comp_l y)\comp_k (x' \comp_l y')).
(x \comp_k x')\comp_l (y \comp_k y')=(x \comp_l y)\comp_k (x' \comp_l y').
\]
\end{description}
We will use the same letter to denote an $\oo$-category and its underlying $\oo$\nbd-magma. A \emph{morphism of $\oo$-categories} (or \emph{$\oo$-functor}), $f : X \to Y$, is simply a morphism of the underlying $\oo$\nbd-magmas. We denote by $\oo\Cat$ the category of $\oo$-categories and morphisms of $\oo$-categories. This category is clearly locally presentable.
......@@ -739,11 +739,11 @@ The following proposition is the key result of this section. It is slightly less
Concretely, Proposition \ref{prop:freeonpolygraph} gives us a recipe to construct free $\oo$-categories. It suffices to give a formal list of generating cells of the form:
\begin{itemize}[label=-]
\item generating $0$-cells : $x^0, y^0, \dots$
\item generating $1$-cells : $x^1 : \sigma(x^1) \to \tau(x^1): y^1 : \sigma(y^1) \to \tau(y^1), \dots$
\item generating $2$-cells : $x^2 : \sigma(x^2) \to \tau(x^2): y^2 : \sigma(y^2) \to \tau(y^2), \dots$
\item generating $1$-cells : $x^1 : \sigma(x^1) \to \tau(x^1), y^1 : \sigma(y^1) \to \tau(y^1), \dots$
\item generating $2$-cells : $x^2 : \sigma(x^2) \to \tau(x^2), y^2 : \sigma(y^2) \to \tau(y^2), \dots$
\item $\dots$,
\end{itemize}
where for a generating $k$-cell $x$ with $k>0$, $\sigma(x)$ and $\tau(x)$ are $(k-1)$-cells of the free $(k-1)$-category recursively generated by the generating cells of dimension strictly lower than $k$.
where for a generating $k$-cell $x$ with $k>0$, $\sigma(x)$ and $\tau(x)$ are parallel $(k-1)$-cells of the free $(k-1)$-category recursively generated by the generating cells of dimension strictly lower than $k$.
\end{paragr}
\begin{example}
The data of $1$-cellular extension $\E$ is nothing but the data of a graph $G$ (or $1$-graph in the terminology of of \ref{paragr:defncat}), and it is not hard to see that, in that case, $\E^*$ is nothing but the free category on $G$. That is to say, the category whose objects are those of $G$ and whose arrows are strings of composable arrows of $G$; the composition being given by concatenation of strings. Hence, from Proposition \ref{prop:freeonpolygraph}, a ($1$-)category is free in the sense of Definition \ref{def:freeoocat} if and only if it is (isomorphic to) a free category on a graph.
......@@ -771,3 +771,130 @@ The following proposition is the key result of this section. It is slightly less
\end{example}
\section{Cells of free $\oo$-categories as words}
In this section, we undertake to give a more explicit construction of the $(n+1)$\nbd-category $\E^*$ generated by an $n$-cellular cellular extension $\E=(C,\Sigma,\sigma,\tau)$. By definition of $\E^*$, this amounts to give an explicit description of a particular type of colimit in $\oo\Cat$. Note also that since $\tau_{\leq n}(\E^*)=C$, all we need to do is to describe the $(n+1)$-cells of $\E^*$. This will take place in two steps: first we construct what ought to be called the \emph{free $(n+1)$-magma generated by $\E$}, for which the $(n+1)$-cells are really easy to describe, and then we quotient out these cells as to obtain an $(n+1)$-category, which will be $\E^*$.
Recall that an $n$-category is a particular case of $n$-magma.
\begin{paragr}\label{paragr:defwords}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. We denote by $\W[\E]$ the set of finite words of the alphabet that has:
\begin{itemize}[label=-]
\item a symbol $\cc_{\alpha}$ for each $\alpha \in \Sigma$,
\item a symbol $\ii_{x}$ for each $x \in C_n$,
\item a symbol $\fcomp_k$ for each $0 \leq k \leq n$,
\item a symbol of opening parenthesis $($,
\item a symbol of closing parenthesis $)$.
\end{itemize}
If $w$ and $w'$ are two elements of $\W[\E]$, we write $ww'$ for their concatenation. We now define the subset $\T[\E] \subseteq \W[\E]$ of \emph{well formed words} (or \emph{terms}) on the previous alphabet together with two maps $\src, \trgt : \T[E] \to C_n$ in the following recursive way:
\begin{itemize}[label=-]
\item for every $\alpha \in \Sigma$, the word $(\cc_{\alpha})$ is well formed and we have
\[
\src((\cc_{\alpha}))=\sigma(\alpha) \text{ and } \trgt((\cc_{\alpha}))=\tau(\alpha),
\]
\item for every $x \in C_n$, the word $(\ii_{x})$ is well formed and we have
\[
\src((\ii_{x}))=\trgt((\ii_{x}))=x,
\]
\item if $w$ and $w'$ are well formed words such that $\src(w)=\trgt(w')$, then the word $(w\fcomp_n w')$ is well formed and we have
\[
\src((w\fcomp_n w'))=s(w') \text{ and } \trgt((w\fcomp_n w'))=t(w),
\]
\item if $w$ and $w'$ are well formed words such that $\src_k(\src(w))=\trgt_k(\trgt(w))$ for some $0 \leq k < n$, then the word $(w\fcomp_k w')$ is well formed and we have
\[
\src((w \fcomp_k w'))=\src(w)\comp_k s(w') \text{ and } \trgt((w \fcomp_k w'))=\trgt(w) \comp_k \trgt(w').
\]
\end{itemize}
As usual, for $0\leq k<n$, we define $\src_k,\trgt_k : \T[\E] \to C_k$ to be respectively the iterated source and target (and we set $\src_n=\src$ and $\trgt_n=\trgt$ for consistency).
\end{paragr}
\begin{paragr}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. Let $\E^{+}$ denote the $(n+1)$-magma defined in the following fashion:
\begin{itemize}[label=-]
\item for every $0 \leq k \leq n$, we have $(\E^+)_k:=C_k$; the source, target, compositions of $k$-cells for $0 < k \leq n$ and units on $k$-cells for $0 \leq k <n$ are the ones of $C$,
\item $(\E^{+})_{n+1}=\T[\E]$,
\item the source and target maps $\src,\trgt : (\E^{+})_{n+1} \to (\E^{+})_{n}$ are the ones defined in \ref{paragr:defwords},
\item for every $n$-cell $x$, the unit on $x$ is given by the word
\[
(\ii_x),
\]
\item for $0 \leq k \leq n$, the $k$-composition of two $(n+1)$-cells $w$ and $w'$ such that $\src_k(w)=\trgt_k(w')$ is given by the word
\[
(w\fcomp_k w').
\]
\end{itemize}
Hence, by definition, $\E^+$ satisfy all the axioms of $\oo$-categories up to dimension $n$ only. On the other hand, the $(n+1)$-cells of $\E^+$ make it as far as possible as being a $(n+1)$-category as \emph{none} of the axioms of $\oo$-categories are satisfied for cells of dimension $n+1$.
\end{paragr}
%We would like now to quotient the set of $(n+1)$-cells of $\E^+$ as to make it an $(n+1)$-category. In order to do that, we introduce the notion of congruence.
In the following definition, we consider that a binary relation $\R$ on a set $E$ is nothing but a subset of $E \times E$, and we write $x \; \R \; x'$ to say $(x,x') \in \R$.
\begin{definition}\label{def:congruence}
Let $n\geq 1$. A \emph{congruence} on an $n$-magma $X$ is a binary relation $\R$ on the set of $n$-cells $X_n$ such that:
\begin{enumerate}[label=(\alph*)]
\item $\R$ is an equivalence relation,
\item if $x\; \R\; x'$ then $x$ and $x'$ are parallel,
\item if $x \;\R\; x'$, $y \;\R\; y'$ and if $x$ and $x'$ are $k$-composable for some $0 \leq k <n$ then
\[
x \comp_k x' \; \R \; y\comp_k y'
\]
(which makes sense since $y$ and $y'$ are $k$-composable by the previous axiom).
\end{enumerate}
\end{definition}
\begin{remark}
Beware that in the previous definition, the relation $\R$ is \emph{only} on the set of cells of dimension $n$.
\end{remark}
\begin{example}
Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$. The binary relation $\R$ on $X_n$ defined as
\[
x\R y \text{ if } F(x)=F(y)
\]
is a congruence.
\end{example}
\begin{paragr}
Let $X$ be an $n$-magma with $n \geq 1$ and $\R$ a congruence on $X$. By the first axiom of Definition \ref{def:congruence}, $\R$ is an equivalence relation and we can consider the quotient set $X_n/\R$. We write $[x]$ for the equivalence class of an $n$-cell $x$ of $X$. From the second axiom of Definition \ref{def:congruence}, we can define unambiguously
\[
\src([x]):=\src(x) \text{ and } \trgt([x]):=\trgt(x),
\]
for $x \in X_n$ and from the third axiom, we can define unambiguously
\[
[x]\comp_k[y] := [x \comp_k y]
\]
for $x$ and $y$ $k$-composable $n$-cells of $X$. Altoghether, this defines an $n$-magma, which we denote by $X/{\R}$, whose set $k$-cells is $X_k$ for $0 \leq k < n$, and $X_n/{\R}$ for $k=n$. The composition, source, target and units of cells of dimension strictly lower than $n$ being those of $X$ and the composition, source and target of $n$-cells being given by the above formulas.
\end{paragr}
\begin{definition}
Let $\R$ be a congruence on an $n$-magma $X$ with $n \geq 1$. We say that $\R$ is \emph{categorical} if it satisfies all four following axioms:
\begin{enumerate}
\item for every $k<n$ and every $n$-cell $x$ of $X$, we have
\[
\1^{n}_{\trgt_k(x)}\comp_kx\; \R \; x \text{ and } x \: \R \; x \comp_k\1^n_{\src_k(x)},
\]
\item for every $k<n$ and for all $k$-composable $n$-cells $x$ and $y$ of $X$, we have
\[
1_{x\comp_k y} \; \R \; 1_{x}\comp_k 1_{y},
\]
\item for every $k<n$, for all $n$-cells $x, y$ and $z$ of $X$ such that $x$ and $y$ are $k$-composable, and $y$ and $z$ are $k$-composable, we have
\[
(x\comp_{k}y)\comp_{k}z \; \R \; x\comp_k(y\comp_kz),
\]
\item for all $k,l \in \mathbb{N}$ with $k<n$ and $l<n$, for all $n$-cells $x,x',y$ and $y'$ of $X$ such that
\begin{itemize}
\item[-] $x$ and $y$ are $l$-composable, $x'$ and $y'$ are $l$-composable,
\item[-] $x$ and $x'$ are $k$-composable, $y$ and $y'$ are $k$-composable,
\end{itemize}
we have
\[
(x \comp_k x')\comp_l (y \comp_k y')\; \R \; (x \comp_l y)\comp_k (x' \comp_l y').
\]
\end{enumerate}
\end{definition}
\begin{example} Let $C$ be an $n$-category with $n>1$, which we consider as an $n$-magma. The equality on the set of $n$-cells of $C$, is, by definition, categorical.
\end{example}
\begin{example}
Another important example of categorical congruence is the following. Let $F : X \to Y$ be a morphism of $n$-magmas with $n>1$ and suppose that $Y$ is an $n$-category. Then the binary relation $\R$ on $X_n$ defined as
\[
x \R y \text{ if } F(x)=F(y)
\]
is obviously a \emph{categorical} congruence.
\end{example}
In the following lemma, we use the notation $\tau_{\leq n}^s(X)$ for an $n$-mag
\begin{lemma}
Let $X$ be an $n$-magma with $n>1$ and c
\end{lemma}
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