Commit ceaf1035 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

leaving office

parent 0c0884e9
......@@ -205,7 +205,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\]
is $C$ and the universal arrow $\sk_{n}(C) \to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n} \dashv \iota$.
\end{lemma}
\begin{paragr}
\begin{paragr}\label{paragr:defglobe}
For $n \in \mathbb{N}$, the \emph{$n$-globe} $\sD_n$ is the $n$-category that has:
\begin{itemize}
\item exactly one non-trivial $n$-cell, which we refer to as the \emph{principal $n$-cell} of $\sD_n$, and which we denote by $e_n$,
......@@ -305,8 +305,11 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\]
is cocartesian.
\end{definition}
\begin{remark}
Note that since for all $n<m$, we have $\sk_n \circ \sk_m = \sk_n$, an $\oo$-category $C$ has an $n$-basis if and only if $\sk_n(C)$ has an $n$-basis.
\end{remark}
\begin{paragr}\label{paragr:defnbasisdetailed}
Unfolding the previous definition gives that $E$ is an $n$-basis of $C$ if for every $(n-1)$-category $D$, every $(n-1)$-functor
Unfolding Definition \ref{def:nbasis} gives that $E$ is an $n$-basis of $C$ if for every $(n-1)$-category $D$, every $(n-1)$-functor
\[
F : \tau_{\leq n-1}^{s}(C) \to \tau_{\leq n-1}^{s}(D),
\]
......@@ -326,6 +329,10 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
Intuitively speaking, this means that $\sk_{n}(C)$ has been obtained from $\sk_{n-1}(C)$ by freeling adjoining the cells in $E$.
\end{paragr}
\begin{example}\label{zerobase}
Every $\oo$-category $C$ has a $0$-base which is $C_0$ itself. This is the unique $0$-base possible.
\end{example}
\begin{example}\label{dummyexample}
An $n$-category (seen as an $\oo$-category) always has a $k$-basis for every $k>n$, namely the empty set. Less trivial examples will come along soon.
\end{example}
......@@ -375,7 +382,7 @@ We can now state the promised result, whose proof can be found in \cite[Section
We often say refer to the elements of the $n$-basis of a free $\oo$-category as the \emph{generating $n$-cells}. This sometimes leads to use the alternative terminology \emph{set of generating $n$-cells} instead of \emph{$n$-basis}.
\end{paragr}
So far, we have not yet seen examples of free $\oo$-categories. In order to do that, we will explain in a further section a recursive way of constructing free $\oo$-categories but first we take a little detour.
So far, we have not yet seen examples of free $\oo$-categories. In order to do that, we will explain in a further section a recursive way of constructing free $\oo$-categories; but first we take a little detour.
\section{Suspension of monoids and counting generators}
\begin{paragr}\label{paragr:suspmonoid}
......@@ -436,10 +443,10 @@ This construction will turn out to be of great use many times in this dissertati
is bijective.
\end{lemma}
\begin{proof}
This is an immediate consequence of Lemma \ref{lemma:nfunctortomonoid} and the universal property of an $n$-basis (as explained in Paragraph \ref{paragr:defnbasisdetailed})
This is an immediate consequence of the universal property of an $n$-basis (as explained in Paragraph \ref{paragr:defnbasisdetailed})
\end{proof}
We can now prove the important proposition below.
\begin{proposition}
\begin{proposition}\label{prop:countingfunction}
Let $C$ be an $\oo$-category, and suppose that $C$ has a $n$-basis $E$ with $n\geq 0$. For every $\alpha \in E$, there exists a unique function
\[
w_{\alpha} : C_n \to \mathbb{N}
......@@ -474,13 +481,142 @@ Furthermore, this function satisfies the condition
\[
w_{\alpha}(1_x)=0.
\]
Hence, from Lemma \ref{lemma:nfunctortomonoid} we know that $w_{\alpha}$ necessarily comes from an $n$\nbd-functor $C \to B^n\mathbb{N}$ and the uniqueness follows then from conditions (a) and (b) and Lemma \ref{lemma:freencattomonoid}.
Hence, we can apply Lemma \ref{lemma:nfunctortomonoid} which shows that $w_{\alpha}$ necessarily comes from an $n$\nbd-functor $C \to B^n\mathbb{N}$. Then, the uniqueness follows from conditions (a) and (b) and Lemma \ref{lemma:freencattomonoid}.
\end{proof}
\begin{paragr}
Let $C$ be as in the previous corollary and let $x$ be an $n$-cell of $C$. We sometimes refer to $w_{\alpha}(x)$ as the \emph{weight of $\alpha$ in $x$}. Intuitively speaking, the function $w_{\alpha}$ counts how many times $\alpha$ ``appears'' in $x$. We shall give in a later section an more solid explanation for this intuition.
Let $C$ be an $n$-category with an $n$-basis $E$. For an $n$-cell $x$ of $C$, we refer to the integer $w_{\alpha}(x)$ as the \emph{weight of $\alpha$ in $x$}. The reason for such a name will become clearer later once we give an explicit construction of $w_{\alpha}$ as a function that ``counts the number of occurences of $\alpha$ in an $n$-cell''.
For later reference, let us also highlight the fact that in the proof of the previous proposition, we have shown the important property that if $n>0$, then for $y \in C_{n-1}$, we have
\[
w_{\alpha}(1_y)=0.
\]
This implies that for $n>1$, there might be $n$-cells $x$ such that
\[
x \neq \alpha \text{ and }w_{\alpha}(x)=1.
\]
Indeed, suppose that there exists a $k$-cell $z$ with $0<k<n-1$ which is not a unit on a lower dimensional cell and such that $\trgt_{k-1}(z)=\src_{k-1}(\alpha)$, then we have
\[
w_{\alpha}(\alpha\comp_{k-1}\1^n_z)=w_{\alpha}(\alpha)+w_{\alpha}(\1^n_z)=1.
\]
\end{paragr}
\section{Recursive construction of free $\oo$-categories}
\begin{definition}\label{def:cellularextension}
Let $n \in \mathbb{N}$. A \emph{$n$-cellular extension} consists of a quadruplet $(C,\Sigma,\sigma,\tau)$ where:
\begin{itemize}[label=-]
\item $C$ is an $n$-category,
\item $\Sigma$ is a set,
\item $\sigma$ and $\tau$ are maps $\Sigma \to C_n$ such that for every element $x \in \Sigma$, the $n$-cells $\sigma(x)$ and $\tau(x)$ are parallel.
\end{itemize}
\end{definition}
\begin{paragr}
If we are given an $n$-category $C$, then we also say \emph{a cellular extension of $C$} for a cellular extension of the form $(C,\Sigma,\sigma,\tau)$.
Note also that, while not explicitely
Intuitively speaking, the set $\Sigma$ is a set of formal extra $(n+1)$-cells of $C$. The source and target of every $x \in \Sigma$ are given respectively by $\sigma(x)$ and $\tau(x)$ (which makes sense since these two $n$-cells are parallel). This is why we often adopt the notation
\[
x : a \to b
\]
for an element of $\Sigma$ such that $\sigma(x)=a$ and $\tau(x)=b$.
\end{paragr}
\begin{definition}
Let $\E=(C,\Sigma,\sigma,\tau)$ and $\E'=(C',\Sigma',\sigma',\tau')$ be two $n$-cellular extensions. A morphism of $n$-cellular extensions $\E \to \E'$ consists of a pair $(F,\varphi)$ where:
\begin{itemize}[label=-]
\item F is an $n$-functor $C \to C'$,
\item $\varphi$ is a map $\Sigma \to \Sigma'$,
\end{itemize}
such that for every $x \in \Sigma$, we have
\[
\sigma'(\varphi(x))=F(\sigma(x)) \text{ and } \tau'(\varphi(x))=F(\tau(x)).
\]
\end{definition}
\begin{paragr}
We denote by $n\Cat^{+}$ the category of $n$-cellular extensions and morphisms of $n$-cellular extensions. Note that every $(n+1)$-category $C$ canonically defines an $n$-cellular extension $(\tau^s_{\leq n }(C),C_{n+1},\src,\tau)$ where $\src,\tau : C_{n+1} \to C_n$ 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},s,t).
\end{align*}
\end{paragr}
The following proposition is the key result of this section. It is slightly less trivial than it appears.
\begin{proposition}
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:
\[
\begin{tikzcd}[column sep=huge, row sep=huge]
\displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \sigma(x),\tau(x)\rangle_{x \in \Sigma}}"] & C \ar[d] \\
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\E^*
\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]
\end{tikzcd}.
\]
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
\[
\begin{tikzcd}
\Sigma \ar[d,hook] \ar[r,"\varphi"] & \Sigma' \ar[d,hook] \\
(\E^*)_{n+1} \ar[r,"(G^*)_{n+1}"]&(\E'^*)_{n+1}
\end{tikzcd}
\]
is commutative.
\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 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
\[
\begin{tikzcd}[column sep=huge, row sep=huge]
\displaystyle\coprod_{x \in \Sigma}\sS^n \ar[d,"\displaystyle\coprod_{x \in \Sigma}i_{n+1}"']\ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in \Sigma}}"] & \sk_{n}(\E^*) \ar[d,hook] \\
\displaystyle\coprod_{x \in \Sigma}\sD_{n+1}\ar[r]&\sk_{n+1}(\E^*)
\ar[from=1-1,to=2-2,very near end,phantom,"\ulcorner"]
\end{tikzcd}
\]
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 $\E^*$.
\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
\[
\E_E =(\tau_{\leq n}^s(C),E,\src,\trgt),
\]
where $\src$ and $\trgt$ are simply the restriction to $E$ of the source and target maps $C_{n+1} \to C_n$. The canonical inclusion $E \hookrightarrow C_{n+1}$ induces a morphism of $n$-cellular extensions
\[
\E_E \to U_n(C),
\]
and then, by adjunction, an $(n+1)$-functor
\[
\E_E^* \to C.
\]
\end{paragr}
\begin{proposition}
Let $C$ be an $(n+1)$-category. A subset $E \subseteq C_{n+1}$ is an $(n+1)$-basis of $C$ if and only if the canonical $(n+1)$-functor
\[
\E_E^* \to C
\]
is an isomorphism.
\end{proposition}
\begin{example}
Recall that for a graph $G$ (or $1$-graph in the terminology of \ref{paragr:defncat}), the free category on $G$ is 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.
......@@ -498,4 +634,3 @@ Furthermore, this function satisfies the condition
\]
is not free on a $2$-graph. The reason is that the source (resp. the target) of $\alpha$ is $g \comp_0 f$ (resp. $i \comp_0 h$) which are not generating $1$-cells.
\end{example}
\section{Counting generators}
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