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 ... ... @@ -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 $nn$, 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
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