Commit 8ff3d209 authored by Leonard Guetta's avatar Leonard Guetta
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I maybe need to move up the beginning of section 1.3 to a section on the...

I maybe need to move up the beginning of section 1.3 to a section on the Eckmann-Hilton argument (maybe between actual section 1.1 and 1.2)
parent abc3b090
......@@ -78,6 +78,16 @@ year={2020}
year={2003},
publisher={Springer Netherlands}
}
@article{burroni1993higher,
title={Higher-dimensional word problems with applications to equational logic},
author={Burroni, Albert},
journal={Theor. Comput. Sci.},
volume={115},
number={1},
pages={43--62},
year={1993},
publisher={Citeseer}
}
@article{cisinski2003images,
title={Images directes cohomologiques dans les cat{\'e}gories de modeles},
author={Cisinski, Denis-Charles},
......@@ -274,6 +284,16 @@ note={In preparation}
year={2004},
publisher={International Press of Boston}
}
@article{street1976limits,
title={Limits indexed by category-valued 2-functors},
author={Street, Ross},
journal={Journal of Pure and Applied Algebra},
volume={8},
number={2},
pages={149--181},
year={1976},
publisher={Elsevier}
}
@article{street1987algebra,
title={The algebra of oriented simplexes},
author={Street, Ross},
......
......@@ -150,7 +150,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
((x \comp_k x')\comp_l (y \comp_k y'))=((x \comp_l y)\comp_k (x' \comp_l y')).
\]
\end{description}
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.
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 $1\Cat$.
......@@ -194,9 +194,215 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
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 $C$ there is a canonical filtration
For every $\oo$-category $C$ and $n\geq 0$, we have a canonical inclusion
\[
\sk_{0}(C) \to \sk_{1}(C) \to \cdots \to \sk_{n}(C) \to \cdots,
\sk_{n}(C) \hookrightarrow \sk_{n+1}(C),
\]
which induces 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}
\begin{paragr}
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 $e_n$,
\item exactly two non-trivial $k$-cells for $k<n$; these $k$-cells being parallel and given by the $k$-source and the $k$-target of $e_n$.
\end{itemize}
This completely describes the $n$-category $\sD_n$ as no non-trivial composition can occur. Here are pictures in low dimension:
\[
\sD_0= \begin{tikzcd}\bullet\end{tikzcd},
\]
\[
\sD_1 = \begin{tikzcd} \bullet \ar[r] &\bullet \end{tikzcd},
\]
\[
\sD_2 = \begin{tikzcd}
\bullet \ar[r,bend left=50,""{name = U,below}] \ar[r,bend right=50,""{name=D}]&\bullet \ar[Rightarrow, from=U,to=D]
\end{tikzcd},
\]
\[
\sD_3 = \begin{tikzcd}
\bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]&\bullet \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}]
\arrow[phantom,"\Rrightarrow",from=L,to=R]
\end{tikzcd}.
\]
For any $\oo$-category $C$, the map
\begin{align*}
\Hom_{\oo\Cat}(\sD_n,C) &\to C_n \\
F &\mapsto F(e_n)
\end{align*}
is a bijection, natural in $C$. In other words, the $n$-globe represents the functor
\begin{align*}
\oo\Cat &\to \Set\\
C &\mapsto C_n.
\end{align*}
For an $n$-cell $x$ of $C$, we denote by
\[
\langle x \rangle : \sD_n \to C
\]
the canonically associated $\oo$-functor.
\end{paragr}
\begin{paragr}
For $n \in \mathbb{N}$, the $n$-sphere $\sS^n$ is the $n$-category such for every $k\leq n$, it has exactly two parallel $k$-cells. In other words, we have
\[
\sS^{n}=\sk_n(\sD_{n+1}),
\]
and in particular, we have a canonical inclusion functor
\[
i_{n+1} : \sS^{n} \to \sD_{n+1}.
\]
It is also customary to define $\sS^{-1}$ to be the empty $\oo$-category and $i_{-1}$ to be the unique $\oo$-functor
\[
\emptyset \to \sD_0.
\]
With this definition of $\sS^{-1}$, we can also give a recursive definition of the $n$-sphere for $n\geq 0$ as the following amalgamated sum
\[
\begin{tikzcd}
\sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d]\\
\sD_n \ar[r] & \sS_{n}.
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\]
The two anonymous arrows of the previous square are representing each one of the two parallel $n$-cells of $\sS_n$.
Here are some pictures of the $n$-spheres in low dimension:
\[
\sS_0= \begin{tikzcd}\bullet & \bullet \end{tikzcd},
\]
\[
\sS_1 = \begin{tikzcd} \bullet \ar[r,bend left=50,""{name = U,below}] \ar[r,bend right=50,""{name=D}]&\bullet \end{tikzcd},
\]
\[
\sS_2 = \begin{tikzcd}
\bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]&\bullet \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}]
\end{tikzcd}.
\]
For an $\oo$-category $C$ and $n\geq 0$, an $\oo$-functor
\[
\sS_n \to C
\]
amounts to the data of two parallel $n$-cells of $C$. In other words, $\sS_{n}$ represents the functor $\oo\Cat \to \Set$ that sends an $\oo$-category to the set of its parallel $n$-cells. For $(x,y)$ a pair of parallel $n$-cells of $C$, we denote by
\[
\langle x,y \rangle : \sS_{n} \to C
\]
the canonically associated $\oo$-functor. For example, the $\oo$-functor $i_n$ is nothing but
\[
\langle \src(e_{n+1}),\trgt(e_{n+1}) \rangle : \sS^{n} \to \sD_{n+1}.
\]
\end{paragr}
\section{Free $\oo$-categories}
\begin{definition}
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
\[
\begin{tikzcd}[column sep=huge, row sep=huge]
\displaystyle \coprod_{ x \in E}\ar[d,"\displaystyle\coprod_{x \in E} i_n"'] \sS^{n-1} \ar[r,"{\langle \src(x),\trgt(x)\rangle_{x \in E}}"] & \sk_{n-1}(C) \ar[d,hook] \\
\displaystyle \coprod_{x \in E} \sD_n \ar[r,"\langle x \rangle_{x \in E}"] & \sk_{n}(C)
\end{tikzcd}
\]
is cocartesian.
\end{definition}
\begin{paragr}
Unfolding the previous definition gives that $E$ is an $n$-basis of $C$ if for every $(n-1)$-category $D$, every $(n-1)$-functor
\[
F : \tau_{\leq n-1}^{s}(C) \to \tau_{\leq n-1}^{s}D,
\]
and every map
\[
f : E \to D_n
\]
such that for every $x \in E$,
\[
\src(f(x))=F(\src(x)) \text{ and } \trgt(f(x))=F(\trgt(x)),
\]
there exists a \emph{unique} $n$-functor
\[
\tilde{F} : \tau^{s}_{\leq n}(C) \to D
\]
such that $\tilde{F}_k = F_k$ for every $k<n$ and $\tilde{F}_n(x) = f(x)$ for every $x \in E$.
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{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}
\begin{definition}\label{def:freeoocat}
An $\oo$-category is \emph{free}\footnote{Other common terminology for ``free $\oo$-category'' is ``$\oo$\nbd-category free on a polygraph'' \cite{burroni1993higher} or ``$\oo$\nbd-category free on a computad'' \cite{street1976limits,makkai2005word}.} if it has $n$-basis for every $n \geq 0$.
\end{definition}
\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.
\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*)]
\item $x$ is not a unit on a lower dimensional cell,
\item if $x$ is of the form
\[
x=a\comp_k b
\]
with $k<n$, then either
\[
a=\1^n_{\trgt_k(x)},
\]
or
\[
b=\1^n_{\src_k(x)}.
\]
\end{enumerate}
For $n=0$, all $0$-cells are, by convention, indecomposable.
\end{definition}
We can now state the promised result, whose proof can be found in \cite[Section 4, Proposition 8.3]{makkai2005word}.
\begin{proposition}[Makkai]\label{prop:uniquebasis}
Let $C$ be a free $\oo$-category. For every $n \in \mathbb{N}$, $C$ has a \emph{unique} $n$-basis. The cells of this $n$-basis are exactly the indecomposable $n$-cells of $C$.
\end{proposition}
\begin{remark}
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
\[
(\Sigma_n)_{n \in \mathbb{N}}
\]
where each $\Sigma_n$ is the $n$-basis of $C$.
We often say refer to the elements of the $n$-basis of a free $\oo$-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}.
\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.
A $1$-category $C$ is free in the sense of Definition \ref{def:freeoocat} if and only if there exists a graph $G$ such that $C$ is (isomorphic) to the free category on $G$. The generating $1$-cells of $G$ are given by the strings of length $1$.
\end{example}
\begin{example}
The notion of free category on a graph is easily generalized to the notion of free $n$-category on a $n$-graph (with $n \in \mathbb{N}\cup\{\infty\}$). As in the previous example, any free $n$-category on an $n$-graph is free in the sense of Definition \ref{def:freeoocat}. \emph{However}, the converse is not true for $n>1$. The point is that in a free $n$-category on an $n$-graph, the source and target of a $k$\nbd-generating cell must be $(k-1)$-generating cells; whereas for a free $n$\nbd-category, they can be any $(k-1)$-cells (not necessarily generating). For example, the free $2$-category pictured as
\[
\begin{tikzcd}
& B\ar[rd,"g"] & \\
A\ar[ru,"f"] \ar[rd,"h"'] & & C \\
& D\ar[ru,"i"'] &
\ar[from=3-2,to=1-2,Rightarrow,shorten <= 1.5em, shorten >= 1.5em, "\alpha"]
\end{tikzcd}
\]
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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