Commit 0c0884e9 authored by Leonard Guetta's avatar Leonard Guetta
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Ouf, Proposition 1.3.5 is done. I need to wrap up this section with somme...

Ouf, Proposition 1.3.5 is done. I need to wrap up this section with somme additional commentary on the weight of units and the name 'counting function'. I then need to move Examples 1.3.7 and 1.3.8 in the next section on inductive definition of free oo-categories
parent 8ff3d209
......@@ -153,7 +153,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
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.
\end{paragr}
\begin{paragr}\label{paragr:defncat}
For $n \in \mathbb{N}$, the notions of \emph{$n$-graph}, \emph{$n$-magma} and \emph{$n$-category} are defined as truncated version of $\oo$-graph, $\oo$-magma, and $\oo$-category in an obvious way. For example, a $1$-category is nothing but a usual (small) category. The category of $n$-categories and morphisms of $n$-categories (or $n$-functors) is denoted by $n\Cat$. When $n=1$, we also use the notation $\Cat$ instead of $1\Cat$.
For $n \in \mathbb{N}$, the notions of \emph{$n$-graph}, \emph{$n$-magma} and \emph{$n$-category} are defined as truncated version of $\oo$-graph, $\oo$-magma, and $\oo$-category in an obvious way. For example, a $0$-category is a set and a $1$-category is nothing but a usual (small) category. The category of $n$-categories and morphisms of $n$-categories (or $n$-functors) is denoted by $n\Cat$. When $n=0$ and $n=1$, we most often use the notation $\Set$ and $\Cat$ instead of $0\Cat$ and $1\Cat$.
For every $n\geq 0$, there is a canonical functor
\[
......@@ -186,24 +186,25 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}.
The functor $\iota$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units on lower dimensional cells.
\end{paragr}
\begin{paragr}
For $n \geq 0$, we define the $n$-skeleton functor $\sk_n : \oo\Cat \to \oo\Cat$ as
\[
\sk_n := \iota \circ \tau^{s}_{\leq n}.
\]
This functor preserves both limits and colimits.
% and, for consistency in later definitions, we also define $\sk_{(-1)} : \oo\Cat \to \oo\Cat$ to be the constant functor with value the empty $\oo$-category.\footnote{which is a $(-1)$-category !}
For every $\oo$-category $C$ and $n\geq 0$, we have a canonical inclusion
\[
\sk_{n}(C) \hookrightarrow \sk_{n+1}(C),
\]
which induces a canonical filtration
This functor preserves both limits and colimits. For an $\oo$-category $C$, $\sk_n(C)$ is the sub-$\oo$-category of $C$ generated by the $k$-cells of $C$ with $k\leq n$, in an obvious sense. In particular, we have a canonical filtration
\[
\sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots,
\]
whose colimit is $C$; the universal arrow $\sk_{n}(C) \to C$ being given by the co-unit of the adjunction $\tau^{s}_{\leq n} \dashv \iota$.
\end{paragr}
induced by the inclusion. We leave the proof of the following lemma as an easy exercice to the reader.
\end{paragr}
\begin{lemma}\label{lemma:filtration}
Let $C$ be an $\oo$-category. The colimit of the canonical filtration
\[
\sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots
\]
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}
For $n \in \mathbb{N}$, the \emph{$n$-globe} $\sD_n$ is the $n$-category that has:
\begin{itemize}
......@@ -294,7 +295,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\]
\end{paragr}
\section{Free $\oo$-categories}
\begin{definition}
\begin{definition}\label{def:nbasis}
Let $C$ be an $\oo$-category and $n \geq 0$. A subset $E \subseteq C_n$ of the $n$-cells of $C$ is an \emph{$n$-basis of $C$} if the commutative square
\[
\begin{tikzcd}[column sep=huge, row sep=huge]
......@@ -304,10 +305,10 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\]
is cocartesian.
\end{definition}
\begin{paragr}
\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
\[
F : \tau_{\leq n-1}^{s}(C) \to \tau_{\leq n-1}^{s}D,
F : \tau_{\leq n-1}^{s}(C) \to \tau_{\leq n-1}^{s}(D),
\]
and every map
\[
......@@ -334,7 +335,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\begin{paragr}\label{paragr:freencat}
By considering $n\Cat$ as a subcategory of $\oo\Cat$, the previous definition also works for $n$-categories. It follows from Example \ref{dummyexample} that an $n$-category is free if and only if it has a $k$-basis for every $0 \leq k \leq n$.
\end{paragr}
Before giving examples of free $\oo$-categories, we wish to recall an important result due to Makkai concerning the uniqueness of $n$-basis for a free $\oo$-category. First we need the following definition.
We wish now to recall an important result due to Makkai concerning the uniqueness of $n$-basis for a free $\oo$-category. First we need the following definition.
\begin{definition}
Let $C$ be an $\oo$-category. For $n>0$, an $n$-cell $x$ of $C$ is \emph{indecomposable} if both following conditions are satisfied:
\begin{enumerate}[label=(\alph*)]
......@@ -362,15 +363,124 @@ We can now state the promised result, whose proof can be found in \cite[Section
Beware that there is a subtlety in the previous proposition. It is not in general true that if an $\oo$-category $C$ has an $n$-basis then it is unique. The point is that we need the existence of $k$-basis for $k<n$ in order to prove the uniqueness the $n$-basis. (See the paper of Makkai cited previously for details.)
\end{remark}
\begin{paragr}
Proposition \ref{prop:uniquebasis} allows us to speak of \emph{the} $n$-basis of a free $\oo$-category $C$ and more generally of the \emph{basis} of $C$ for the sequence
Proposition \ref{prop:uniquebasis} allows us to speak of \emph{the} $k$-basis of a free $\oo$-category $C$ and more generally of the \emph{basis} of $C$ for the sequence
\[
(\Sigma_k)_{k \in \mathbb{N}}
\]
where each $\Sigma_k$ is the $k$-basis of $C$. In the case that $C$ is a free $n$-category with $n$ finite and in light of Example \ref{dummyexample}, we will also speak of \emph{the basis of $C$} as the finite sequence
\[
(\Sigma_n)_{n \in \mathbb{N}}
(\Sigma_k)_{0 \leq k \leq n}.
\]
where each $\Sigma_n$ is the $n$-basis of $C$.
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.
\section{Suspension of monoids and counting generators}
\begin{paragr}\label{paragr:suspmonoid}
Let $M$ be a monoid. For any $n >0$, let $B^{n}M$ be the $n$-magma such that:
\begin{itemize}[label=-]
\item it has only one object $\star$,
\item it has only one $k$-cell for $0 < k <n$, which is $\1^k_{\star}$,
\item the set of $n$-cells is (the underlying set of) $M$,
\item for any $k<n$, the $k$-composition of $n$-cells is given by the composition law of the monoid (which makes sense since all $n$-cells are $k$-composable) and the ony unital $n$-cell is given by the neutral element of the monoid.
\end{itemize}
It is sometimes useful to extend the above construction to the case $n=0$ by saying that $B^0M$ is the underlying set of the monoid $M$.
For $n=1$, $B^1M$ is nothing but the monoid $M$ seen as $1$-category.
We often say refer to the elements of the $n$-basis of a free $\oo$-categoryas the \emph{generating $n$-cells}. This sometimes leads to use the alternative terminology \emph{set of generating $n$-cells} instead of \emph{$n$-basis}.
For $n>1$, while it is clear that all first three axioms of $n$-category (units, fonctoriality of units and associativity) hold, it is not always true that the exchange rule is satisfied. If $\ast$ denotes the composition law of the monoid, this axioms states that for all $a,b,c,d \in M$, we must have
\[
(a \ast b) \ast (c \ast d) = (a \ast c ) \ast (b \ast d).
\]
It is straightforward to see that a necessary and sufficient condition for this equation to hold is to require that $M$ be commutative. Hence, we have proven the following lemma.
\end{paragr}
\begin{lemma}
Let $M$ be a monoid and $n \in \mathbb{N}$. Then:
\begin{itemize}
\item if $n=1$, $B^1M$ is a $1$-category,
\item if $n>1$, the $n$-magma $B^nM$ is an $n$-category if and only $M$ is commutative.
\end{itemize}
\end{lemma}
This construction will turn out to be of great use many times in this dissertation and we now explore a few of its properties.
\begin{lemma}\label{lemma:nfunctortomonoid}
Let $C$ be an $n$-category with $n\geq 1$ and $M=(M,\ast,1)$ a monoid (commutative if $n>1$). The map
\begin{align*}
\Hom_{n\Cat}(C,B^nM) &\to \Hom_{\Set}(C_n,M)\\
F &\mapsto F_n
\end{align*}
is injective and its image consists exactly of those functions $f : C_n \to M$ such that:
\begin{itemize}[label=-]
\item for every $0 \leq k <n$ and every pair $(x,y)$ of $k$-composable $n$-cells of $C$, we have
\[
f(x\comp_ky)=f(x)\ast f(y),
\]
\item for every $x \in C_{n-1}$, we have
\[
f(1_x)=1.
\]
\end{itemize}
\end{lemma}
\begin{proof}
The injectivity part follows from the fact that $(B^nM)_k$ is a singleton set for any $k<n$ and hence, an $n$-functor $F : C \to B^nM$ is entirely determined by its restriction to the $n$-cells $F_n : C_n \to M$.
The characterization of the image is immediate once noted that the requirements are only the translation of the axioms of $n$-functor in this particular case.
\end{proof}
\begin{lemma}\label{lemma:freencattomonoid}
Let $C$ be an $n$-category with $n \geq 1$ and $M$ a monoid (commutative if $n>1$). If $C$ has $n$-basis $E$, then the map
\begin{align*}
\Hom_{n\Cat}(C,B^nM) &\to \Hom_{\Set}(E,M)\\
F &\mapsto F_n\vert_{E}
\end{align*}
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})
\end{proof}
We can now prove the important proposition below.
\begin{proposition}
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
\[
w_{\alpha} : C_n \to \mathbb{N}
\]
such that
\begin{enumerate}[label=(\alph*)]
\item $w_{\alpha}(\alpha)=1$,
\item $w_{\alpha}(\beta)=0$ for every $\beta \in E$ such that $\beta \neq \alpha$,
\item for every $0 \leq k<n$ and every pair $(x,y)$ of $k$-composable $n$-cells of $C$, we have
\[
w_{\alpha}(x\comp_k y)=w_{\alpha}(x)+w_{\alpha}(y).
\]
\end{enumerate}
\iffalse
Furthermore, this function satisfies the condition
\begin{enumerate}[label=(\alph*),resume]
\item If $n>0$, for every $x \in C_{n-1}$, we have
\[
w_{\alpha}(1_x)=0.
\]
\end{enumerate}
\fi
\end{proposition}
\begin{proof}
Notice first that $C$ has an $n$-basis if and only if $\sk_n(C)$ has a basis. Hence we can suppose that $C$ is an $n$-category.
For $n=0$, conditions (c) and (d) are vacuous and the assertion is trivial.
Now let $n>0$ and consider the monoid $\mathbb{N}=(\mathbb{N},+,0)$. The existence of a function $C_n \to \mathbb{N}$ satisfying conditions (a) and (b) follows from Lemma \ref{lemma:freencattomonoid} and the fact that it satisfies (c) follows Lemma \ref{lemma:nfunctortomonoid}.
For the uniqueness, notice first that since for any $x \in C_{n-1}$, we have $1_x =1_x\comp_{n-1} 1_x$, condition (c) implies that
\[
w_{\alpha}(1_x)=0.
\]
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}.
\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.
Note also that, while not explicitely
\end{paragr}
We now turn to basic examples of free $\oo$-categories.
\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.
......@@ -389,20 +499,3 @@ We now turn to basic examples of free $\oo$-categories.
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}
\begin{paragr}
Let $M$ be a monoid. For any $n >0$, let $B^{n}M$ be the $n$-magma such that:
\begin{itemize}[label=-]
\item it has only one object $\star$,
\item it has only one $k$-cell for $0 < k <n$, which is $\1^k_{\star}$,
\item the set of $n$-cells is (the underlying set of) $M$,
\item for any $k<n$, the $k$-composition of $n$-cells is given by the composition law of the monoid (which makes sense since all $n$-cells are $k$-composable) and the ony unital $n$-cell is given by the neutral element of the monoid.
\end{itemize}
For $n=1$, $B^1M$ is nothing but the monoid $M$ seen as $1$-category (and hence as a $1$-magma). For $n>1$, while it is clear that all first three axioms of $n$-category (units, fonctoriality of units and associativity) holds, it is not always true that the exchange rule holds. If $\ast$ denotes the composition law of the monoid, this axioms states that for all $a,b,c,d \in M$, we must have
\[
(a \ast b) \ast (c \ast d) = (a \ast c ) \ast (b \ast d).
\]
It is straightforward to see that a necessary and sufficient condition for this equation to hold is to require that $M$ be commutative. Hence we have proven the following lemma.
\end{paragr}
\begin{lemma}
Let $M$ be a monoid and $n>1$. The $n$-magma $B^nM$ is an $n$-category if and only $M$ is commutative. For $n=1$, $B^1M$ is always a $1$-category.
\end{lemma}
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