Commit dcee846c authored by Leonard Guetta's avatar Leonard Guetta
Browse files

dodo

parent 2e48a891
......@@ -610,7 +610,7 @@ Furthermore, this function satisfies the condition
(\E^*)_{n+1} \ar[r,"(G^*)_{n+1}"]&(\E'^*)_{n+1}
\end{tikzcd}
\]
is commutative. Notice also that $j$ is compatible with source and targets in the sense that for every $x \in \Sigma$, we have
is commutative. Notice also that $j$ is compatible with source and target in the sense that for every $x \in \Sigma$, we have
\[
\src(j(x))=\sigma(x) \text{ and } \trgt(j(x))=\tau(x).
\]
......@@ -629,9 +629,14 @@ Furthermore, this function satisfies the condition
\]
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 consequence of the previous lemma, we will often identify $\Sigma$ to a subset of $(\E^*)_{n+1}$. When we do so, it will \emph{always} be via tha map $j$. Note that this identification is compatible with source and target in the sense that the source (resp. target) of $x \in \Sigma$, seen as $(n+1)$-cell of $\E^*$, is exactly $\sigma(x)$ (resp. $\tau(x)$).
\end{paragr}
\iffalse
\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 $(\E^{*})_{n+1}$ and $j$ as the canonical inclusion.
\end{paragr}
\fi
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 $\E^*$.
......@@ -809,9 +814,11 @@ Recall that an $n$-category is a particular case of $n$-magma.
\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).
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{definition}\label{def:sizeword}
The \emph{size} of a well-formed word $w$, denoted by $\vert w \vert$, is the number of symbols $\fcomp_k$ for any $0 \leq k \leq n$ that appear in $w$.
\end{definition}
\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=-]
......@@ -893,8 +900,8 @@ In the following definition, we consider that a binary relation $\R$ on a set $E
\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 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
\begin{example}\label{example:categoricalcongruence}
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
\[
x\; \R \;y \text{ if } F(x)=F(y)
\]
......@@ -902,8 +909,8 @@ In the following definition, we consider that a binary relation $\R$ on a set $E
\end{example}
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}
The following lemma is trivial but nonetheless important. Its immediate proof is ommited.
\begin{lemma}\label{lemma:quotientmagma}
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 how to prove the existence of a congruence defined with a condition such as ``the smallest congruence that contains a given binary relation on the $(n+1)$-cells''.
......@@ -940,6 +947,110 @@ We wish now to see how to prove the existence of a congruence defined with a con
\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}
\iffalse \begin{paragr}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension, $D$ an $(n+1)$-category and
\[
G=(\varphi,F) : \E \to U_n(D)
\]
a morphism of $n$-cellular extensions.
\end{paragr}
\fi
\begin{paragr}
Let $\E=(C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension and consider the injective map $j : \Sigma \to (\E^*)_{n+1}$ constructed in Paragraph \ref{paragr:freecext}. By induction, we define a map
In particular, for $\E$ an $n$-cellular extension, there exists a smallest categorical congruence on $\E^+$, which we denote here by $\equiv$. Applying Lemma \ref{lemma:quotientmagma} gives us that $\E^+/{\equiv}$ is an $(n+1)$-category. The construction \[\E \mapsto \E^+/{\equiv}\] clearly defines a functor $\Cat_n^+ \to (n+1)\Cat$.
\end{paragr}
We can now prove the expected result.
\begin{proposition}
Let $\E$ be an $n$-cellular extension and let $\equiv$ be the smallest categorical congruence on $\E^+$. The $(n+1)$-category $\E^+/{\equiv}$ is naturally isomorphic to $\E^*$.
\end{proposition}
\begin{proof}
The strategy of the proof is to show that the functor $\E \mapsto \E^+/{\equiv}$ is left adjoint to $U_n : (n+1)\Cat \to \Cat_n^+$. The result will follow then from the uniqueness (up to natural isomorphism) of left adjoints.
Let $\E = (C,\Sigma,\sigma,\tau)$ be an $n$-cellular extension, $D$ an $(n+1)$-category and let
\[
G=(\varphi,F) : \E \to U_n(D)
\]
be a morphism of $n$-cellular extensions. We recursively define a map
\[
\overline{\varphi} : \T[\E] \to D_{n+1}
\]
as:
\begin{itemize}[label=-]
\item $\overline{\varphi}(\cc_{\alpha}) = \varphi(\alpha)$ for $\alpha \in \Sigma$,
\item $\overline{\varphi}(\ii_x)=1_{F(x)}$ for $x \in C_n$,
\item $\overline{\varphi}((w \fcomp_k w'))=\overline{\varphi}(w)\comp_k\overline{\varphi}(w')$.
\end{itemize}
A straightforward induction shows that $\overline{\varphi}$ is compatible with source, target and units. Hence, we can define a morphism of $(n+1)$-magmas
\[
\overline{G} : \E^+ \to D
\]
as
\[
\overline{G}_{n+1}=\overline{\varphi}
\]
and
\[
\overline{G}_k=F_k \text{ for }0 \leq k \leq n.
\]
Since $D$ is an $(n+1)$-category, the binary relation $\R$ on $\T[\E]$ defined as
\[
x\; \R \; y \text{ if } \overline{\varphi}(x)=\overline{\varphi}(y)
\]
is a \emph{categorical} congruence on $\E^+$ (Example \ref{example:categoricalcongruence}). If we denote by $\equiv$ the smallest categorical congruence on $\E^+$ then, by definition, $\equiv$ is included in $\R$. In particular, the map $\overline{\varphi}$ induces a map
\[
\widehat{\varphi} : \T[\E]/{\equiv} \to D_{n+1}.
\]
Since $\equiv$ is a congruence, this map is compatible with source and target, and it is straightforward to check that it is also compatible with units. Hence, we have an $(n+1)$-functor
\[
\widehat{G} : \E^+/{\equiv} \to D
\]
defined as
\[
\widehat{G}_{n+1} = \widehat{\varphi}
\]
and
\[
\tau_{\leq n }^s(\widehat{G})=F.
\]
Altogether, we have constructed a map
\begin{align*}
\Hom_{\Cat_n^+}(\E,U_n(D)) &\to \Hom_{(n+1)\Cat}(\E^+/{\equiv},D)\\
G &\mapsto \hat{G}
\end{align*}
which is clearly natural in $\E$ and $D$. What we want to prove is that this map is a bijection.
Let us start with the surjectivity. Let $H : \E^+/{\equiv} \to \D$ be a $(n+1)$-functor. We define a map $\varphi : \Sigma \to D_{n+1}$ as
\begin{align*}
\varphi : \Sigma &\to D_{n+1}\\
\alpha &\mapsto H_{n+1}([\cc_{\alpha}]),
\end{align*}
where $[w]$ is the equivalence class under $\equiv$ of an element $w \in \T[\E]$. All we need to show is that
\[
\widehat{\varphi}=H_{n+1}.
\]
Let $z$ be an element of $\T[\E]/{\equiv}$ and let us choose $w \in \T[\E]$ such that $z=[w]$. We proceed to show that $\widehat{\varphi}(z)=H_{n+1}(z)$ by induction on the size of $w$ (Definition \ref{def:sizeword}).
If $|w|=0$, then either $w=\cc_{\alpha}$ for some $\alpha \in \Sigma$ or $w = \ii_x$ for some $x \in C_n$. In the first case, we have
\[
\widehat{\varphi}([\cc_{\alpha}])=\overline{\varphi}(\cc_{\alpha})=\varphi(\alpha)=H_{n+1}([\cc_{\alpha}]),
\]
and in the second case we have,
\[
\widehat{\varphi}([\ii_x])=\overline{\varphi}(\ii_x)=1_{H_n(x)}=H_{n+1}([\ii_x])
\]
where for the last equality, we used the fact that $[\ii_x]$ is the unit on $x$ in $\E^+/{\equiv}$.
Now if $|w| = n+1$ with $n \geq 0$, then $w=(w'\fcomp_k w'')$ for some $w', w'' \in \T[\E]$ that are $k$-composable with $k\leq n$. Hence, using the induction hypothesis, we have
\begin{align*}
\widehat{\varphi}([w])=\widehat{\varphi}([(w' \fcomp_k w'')])=\widehat{\varphi}([w']\comp_k [w''])&=\widehat{\varphi}([w'])\comp_k\widehat{\varphi}([w''])\\&= H_{n+1}([w'])\comp_kH_{n+1}([w''])\\&=H_{n+1}([w']\comp_k [w''])\\&=H_{n+1}([(w' \fcomp_k w'')])\\&=H_{n+1}([w]).
\end{align*}
We now turn to the injectivity. Let $G=(\varphi,F)$ and $G'=(\varphi',F')$ be two morphism of $n$-cellular extensions $\E \to U_n(D)$ such that $\widehat{G}=\widehat{G'}$. Since $F=\tau_{\leq n}^s(\widehat{G})=\tau_{\leq n}^s(\widehat{G'})=F'$, all we have to show is that
\[
\varphi=\varphi'.
\]
But, by definition, for every $\alpha \in \Sigma$ we have
\[
\varphi(\alpha)=\widehat{\varphi}([\cc_{\alpha}])=\widehat{\varphi'}([\cc_{\alpha}])=\varphi'(\alpha).
\]
\end{proof}
\begin{paragr}
In particular, the previous proposition tells us that $n$-cells in a free $\oo$-category can be represented as a (well formed) words made up of the generating $n$-cells and units on lower dimensionan cells.
\end{paragr}
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