### 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 k0, 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 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 $0 \leq k 1$). 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 $0 \leq k0$, 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 $0 < k 1$, 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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