Commit de644776 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

security commit

parent 498aacaf
......@@ -576,71 +576,84 @@ Furthermore, this function satisfies the condition
\[
\sigma'(\varphi(x))=F(\sigma(x)) \text{ and } \tau'(\varphi(x))=F(\tau(x)).
\]
\end{definition}
\end{definition}
\begin{paragr}
We denote by $n\Cat^{+}$ the category of $n$-cellular extensions and morphisms of $n$-cellular extensions. Every $(n+1)$-category $C$ canonically defines an $n$-cellular extension $(\tau^s_{\leq n }(C),C_{n+1},\src,\trgt)$ where $\src,\trgt : C_{n+1} \to C_n$ are the source and target maps of $C$. This defines a functor
\begin{align*}
U_n : (n+1)\Cat &\to n\Cat^+\\
C &\mapsto (\tau^s_{\leq n }(C),C_{n+1},\src,\trgt).
\end{align*}
\end{paragr}
The following proposition is the key result of this section. It is slightly less trivial than it appears.
\begin{proposition}\label{prop:fromcexttocat}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension and let $\E^*$ be the $(n+1)$-category defined as the following amalgamated sum:
\[
On the other hand, any $\E=(D,\Sigma,\sigma,\tau)$, cellular extension of an $n$-category $D$, yields an $(n+1)$-category $\E^*$ defined as the following amalgamated sum:
\begin{equation}\label{squarefreecext}
\begin{tikzcd}[column sep=huge, row sep=huge]
\displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & C \ar[d] \\
\displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & D \ar[d] \\
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\E^*
\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]
\end{tikzcd}.
\]
Then, $\E^*$ has an $(n+1)$-basis which is isomorphic to $\Sigma$. Moreover, the identification of $\Sigma$ as a subset of $(\E^*)_{n+1}$ is natural in that, for any morphism of $n$-cellular extensions $G=(F,\varphi) : \E \to \E'$, the induced $(n+1)$-functor $G^* : \E^* \to \E^*$ is such that the square
\end{equation}
This actually defines a functor
\begin{align*}
n\Cat^+ &\to (n+1)\Cat \\
\E &\mapsto \E^*,
\end{align*}
which is easily checked to be left adjoint to $U_n$.
Now let $\phi : \coprod_{x \in \Sigma} \sD_{n} \to \E^*$ the bottom map of square \eqref{squarefreecext}. It induces a canonical map
\begin{align*}
j: \Sigma &\to (\E^*)_{n+1}\\
x &\mapsto \phi_x(e_{n+1}),
\end{align*}
where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}). Notice that this map is natural in that, for any morphism of $n$-cellular extensions $G=(F,\varphi) : \E \to \E'$, the square
\[
\begin{tikzcd}
\Sigma \ar[d,hook] \ar[r,"\varphi"] & \Sigma' \ar[d,hook] \\
\Sigma \ar[d,"j"] \ar[r,"\varphi"] & \Sigma' \ar[d,"j'"] \\
(\E^*)_{n+1} \ar[r,"(G^*)_{n+1}"]&(\E'^*)_{n+1}
\end{tikzcd}
\]
is commutative.
\end{proposition}
is commutative. Notice also that $j$ is compatible with source and targets in the sense that for every $x \in \Sigma$, we have
\[
\src(j(x))=\sigma(x) \text{ and } \trgt(j(x))=\tau(x).
\]
\end{paragr}
\begin{lemma}\label{lemma:basisfreecext}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension. The canonical map
\[
j: \Sigma \to (\E^*)_{n+1}
\]
is injective.
\end{lemma}
\begin{proof}
Notice first that since the map $i_{n+1} : \sS^n \to \sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1}) \to \sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to \E^*$ can be identified with the canonical inclusion $\sk_n(\E^*) \to \sk_{n+1}(\E^*)=\E^*$.
A thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ as the amalgamated sum \eqref{squarefreecext} is sufficient enough to prove the existence, for each $x \in \Sigma$, of a function
\[
w_{x} : (\E^*)_{n+1} \to \mathbb{N}
\]
such that $w_{x}(j(x))=1$ and $w_{x}(j(y))=0$ for any $y \in \Sigma$ with $y\neq x$. In particular, this implies that $j$ is injective.
\end{proof}
\begin{paragr}
In particular, the previous lemma tells us that for any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the set of indeterminates $\Sigma$ canonically defines a subset of the $(n+1)$-cells of $\E^*$. As of now, we will consider $\Sigma$ as a subset of $(\E^{*})_{n+1}$.
\end{paragr}
We can now prove the following proposition, which is the key result of this section. It is slightly less trivial than it appears.
\begin{proposition}\label{prop:fromcexttocat}
For any $n$-cellular extension $\E=(C,\Sigma,\sigma,\tau)$, the subset $\Sigma \subseteq (\E^{*})_{n+1}$ is an $(n+1)$-basis of $\E^*$.
\end{proposition}
\begin{proof}
Notice first that since the map $i_{n+1} : \sS^n \to \sD_{n+1}$ is nothing but the canonical inclusion $\sk_{n}(\sD_{n+1}) \to \sk_{n+1}(\sD_{n+1})=\sD_{n+1}$, it follows easily from square \eqref{squarefreecext} and the fact the the skeleton functors preserve colimits that $C$ is canonically isomorphic to $\sk_n(\E^*)$ and that the map $C \to \E^*$ can be identified with the canonical inclusion $\sk_n(\E^*) \to \sk_{n+1}(\E^*)=\E^*$.
Now let $\phi : \coprod_{x \in \Sigma} \sD_{n} \to \E^*$ the bottom map of the cocartesian square of the proposition and consider the map
\begin{align*}
j: \Sigma &\to (\E^*)_{n+1}\\
x &\mapsto \phi_x(e_{n+1}),
\end{align*}
where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}).
The only subtlety of the proposition is to show that $j$ is injective and hence, that $\Sigma$ can be identified to a subset of $(n+1)$-cells of $E^*$. Now a thorough reading of the techniques used in the proofs of Lemma \ref{lemma:nfunctortomonoid}, Lemma \ref{lemma:freencattomonoid} and Proposition \ref{prop:countingfunction} shows that the universal property defining $\E^*$ is sufficient enough to prove the existence, for each $x \in \Sigma$, of a function
\[
w_{x} : (\E^*)_{n+1} \to \mathbb{N}
\]
such that $w_{x}(j(x))=1$ and $w_{x}(j(y))=0$ for any $y \in \Sigma$ with $y\neq x$. In particular, this implies that $j$ is injective.
Altogether, we have shown that there exists a subset of $(\E^*)_{n+1}$ (which we identify with $\Sigma$ via the map $j$) such that the square
where $e_{n+1}$ is the principal $(n+1)$-cell of $\sD_{n+1}$ (\ref{paragr:defglobe}). Hence, cocartesian square \eqref{squarefreecext} can be identified with
\[
\begin{tikzcd}[column sep=huge, row sep=huge]
\displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*)
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*).
\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]
\end{tikzcd}
\]
is cocartesian, which proves the first part of the proposition.
The second part, concerning the naturality of the identification of $\Sigma$ as a subset of $(\E^*)_{n+1}$ is clear from the definition of $j$.
\end{proof}
\begin{corollary}
Let $n \geq 0$. The functor
\begin{align*}
n\Cat^+ &\to (n+1)\Cat \\
\E &\mapsto \E^*
\end{align*}
is left adjoint to the functor $U_n : (n+1)\Cat \to n\Cat^+$.
\end{corollary}
\begin{proof}
This is simply a reformulation of the universal property defining $\E^*$.
Since from $j : \Sigma \to (\E^{*})_{n+1}$ is injective, we can consider $\Sigma$ as a subset of $(\E^{*})_{n+1})$, and then, by definition, $\E^*$ has $\Sigma$ as an $(n+1)$-basis.
\end{proof}
\begin{paragr}
Let $C$ be an $(n+1)$-category and $E$ be a subset $E \subseteq C_{n+1}$. This defines a cellular extension
......@@ -733,7 +746,7 @@ The following proposition is the key result of this section. It is slightly less
The previous proposition admits an obvious truncated version for free $n$-categories with $n$ finite. In that case, we only need a finite sequence $(\E^{(k)}))_{ -1 \leq k \leq n-1}$ of cellular extensions.
\end{remark}
\begin{remark}
The data of a sequence $(\E^{(n)})_{n \geq -1}$ as in Proposition \ref{prop:freeonpolygraph} is commonly referred to in the litterature of the filed as a \emph{computad} \cite{street1976limits} or \emph{polygraph} \cite{burroni1993higher}; and consequently a $\oo$-category which is free in the sense of definition \ref{def:freeoocat} is sometimes referred to as \emph{free on a computad}-or-\emph{polygraph} in the litterature. However, the underlying polygraph of a free $\oo$-category is uniquely determined by the free $\oo$-category itself (a straightforward consequence of Proposition \ref{prop:uniquebasis}), and this is why we chose the shorter terminology \emph{free $\oo$-category}.
The data of a sequence $(\E^{(n)})_{n \geq -1}$ as in Proposition \ref{prop:freeonpolygraph} is commonly referred to in the litterature of the field as a \emph{computad} \cite{street1976limits} or \emph{polygraph} \cite{burroni1993higher}; and consequently a $\oo$-category which is free in the sense of definition \ref{def:freeoocat} is sometimes referred to as \emph{free on a computad}-or-\emph{polygraph} in the litterature. However, the underlying polygraph of a free $\oo$-category is uniquely determined by the free $\oo$-category itself (a straightforward consequence of Proposition \ref{prop:uniquebasis}), and this is why we chose the shorter terminology \emph{free $\oo$-category}.
\end{remark}
\begin{paragr}
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:
......@@ -859,7 +872,7 @@ In the following definition, we consider that a binary relation $\R$ on a set $E
\]
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}
\begin{definition}\label{def:categoricalcongruence}
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
......@@ -885,16 +898,55 @@ In the following definition, we consider that a binary relation $\R$ on a set $E
\]
\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.
\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
Another important example 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
\[
x \R y \text{ if } F(x)=F(y)
x\; \R \;y \text{ if } F(x)=F(y)
\]
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
Similarly to the ``stupid'' truncation of $\oo$-categories (\ref{paragr:defncat}), for an $(n+1)$-magma $X$, we write $\tau_{\leq n}^s(X)$ for the $n$-magma obtained by simply forgetting the cells of dimension $(n+1)$.
The following lemma is trivial but nonetheless important. Its immediate proof is omitted.
\begin{lemma}
Let $X$ be an $n$-magma with $n>1$ and c
Let $X$ be an $n$-magma with $n>1$ and $\R$ a congruence on $X$. If $\tau_{\leq n}^s(X)$ is an $(n-1)$-category and $\R$ is categorical, then $X/{\R}$ is an $n$-category.
\end{lemma}
We wish now to see on which conditions there exists a ``smallest'' congruence on an $n$-magma that contains a given binary relation $\R$ on the set of $n$-cells.
\begin{lemma}\label{lemma:intersectioncongruence}
Let $X$ be an $n$-magma with $n \geq 1$ and $(\R_i)_{i \in I}$ a \emph{non-empty} family of congruences on $X$ (i.e.\ $I$ is not empty). Then, the binary relation
\[
\R:=\bigcap_{i \in I}\R_i
\]
is a congruence.
\end{lemma}
\begin{proof}
The fact that $\R$ satisfies that first and third axiom of Definition \ref{def:congruence} is immediate and do not even require that $I$ be non empty. The only left thing to prove is that if $x \; \R \; x'$, then $x$ and $x'$ are parallel. To see that, notice that since $I$ is not empty, we can choose $i \in I$. Then, by definition, we have $x \; \R_i \; x'$ and thus, $x$ and $x'$ are parallel.
\end{proof}
\begin{remark}
The hypothesis that $I$ be non empty in the previous lemma cannot be ommited because, in this case, we would have
\[
\R=\bigcap_{\emptyset}\R_i = X_n\times X_n,
\]
which means that \emph{all} $n$-cells would be equivalent under $\R$. This certainly does not guarantee that the second axiom of Definition \ref{def:congruence}, i.e.\ that equivalent cells are parallel, is satisfied.
\end{remark}
\begin{lemma}\label{lemma:congruencegenerated}
Let $X$ be an $n$-magma with $n\geq 1$ and $E$ a set of pairs of parallel $n$-cells of $X$. There exists a smallest congruence $\R$ on $X$ such that for every $(x,y)\in E$, we have $x \; \R \;y$.
\end{lemma}
\begin{proof}
Let $I$ be the set of congruence $\mathcal{S}$ on $X$ such that for every $(x,y) \in E$, we have $x\; \mathcal{S} \; y$. All we have to prove is that $I$ is not empty, since in that case, we can apply Lemma \ref{lemma:intersectioncongruence} to the binary relation
\[
\R:=\bigcup_{\mathcal{S} \in I}\S,
\]
which will obviously be the smallest congruence satisfying the desired condition. To see that $I$ is not empty, it suffices to notice that the binary relation ``being parallel $n$-cells'' is a congruence, which obviously is in $I$.
\end{proof}
\begin{proposition}
Let $X$ be an $n$-magma with $n\geq 1$. There exists a smallest categorical congruence on $X$.
\end{proposition}
\begin{proof}
Each four axioms of Definition \ref{def:categoricalcongruence} says that some pairs of parallel $n$-cells must be equivalent under a congruence $\R$ for it to be categorical. The result follows then from Lemma \ref{lemma:congruencegenerated}.
\end{proof}
\begin{paragr}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension
\end{paragr}
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment