Commit abc3b090 authored by Leonard Guetta's avatar Leonard Guetta
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Section 1.1 is over I think. I should start Section 1.2

parent 842fb754
\chapter{Yoga of $\oo$-categories}
\section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories}
\begin{paragr}\label{paragr:defngraph}
\begin{paragr}\label{paragr:defoograph}
An \emph{$\oo$-graph} $X$ consists of an infinite sequence of sets $(X_n)_{n \in \mathbb{N}}$ together with maps
\[ \begin{tikzcd}
X_{n} &\ar[l,"\src",shift left] \ar[l,"\trgt"',shift right] X_{n+1}
......@@ -15,7 +15,9 @@
\end{aligned}
\right.
\end{equation*}
Elements of $X_n$ are called \emph{$n$-cells} or \emph{cells of dimension $n$}. For $n=0$, elements of $X_0$ are also called \emph{objects}. For $x$ a $n$-cell with $n>0$, $\src(x)$ is the \emph{source} of $x$ and $\trgt(x)$ is the \emph{target} of $x$. More generally, for $0\leq k <n$, we define maps $\src_k : X_n \to X_k$ and $\trgt_k : X_n \to X_k$ as
Elements of $X_n$ are called \emph{$n$-cells} or \emph{$m$-arrows} or \emph{cells of dimension $n$}. For $n=0$, elements of $X_0$ are also called \emph{objects}. For $x$ an $n$-cell with $n>0$, $\src(x)$ is the \emph{source} of $x$ and $\trgt(x)$ is the \emph{target} of $x$. We use the notation \[x : a \to b\] to say that $a$ is the source of $x$ and $b$ is the target of $x$.
More generally, for $0\leq k <n$, we define maps $\src_k : X_n \to X_k$ and $\trgt_k : X_n \to X_k$ as
\[
\src_k = \underbrace{\src\circ \dots \circ \src}_{n-k \text{ times}}
\]
......@@ -23,7 +25,7 @@
\[
\trgt_k = \underbrace{\trgt\circ \dots \circ \trgt}_{n-k \text{ times}}.
\]
For a $n$-cell $x$, the $k$-cells $\src_k(x)$ and $\trgt_k(x)$ are respectively the \emph{$k$-source} and the \emph{$k$-target} of $x$.
For an $n$-cell $x$, the $k$-cells $\src_k(x)$ and $\trgt_k(x)$ are respectively the \emph{$k$-source} and the \emph{$k$-target} of $x$.
Two $n$-cells $x$ and $y$ are \emph{parallel} if
\[
......@@ -38,11 +40,163 @@
\[
\src_k(x)=\trgt_k(y).
\]
Note that the expression ``$x$ and $y$ are $k$-composable'' is \emph{not} symetric in $x$ and $y$ and we \emph{should} rather speak of a ``$k$-composable pair $(x,y)$'', although we won't always do it. The set of pairs of $k$-composable $n$-cells is denoted by $C_n\underset{C_k}{\times}C_n$, and is characterized by the following fibred product
Note that the expression ``$x$ and $y$ are $k$-composable'' is \emph{not} symetric in $x$ and $y$ and we \emph{should} rather speak of a ``$k$-composable pair $(x,y)$'', although we won't always do it. The set of pairs of $k$-composable $n$-cells is denoted by $X_n\underset{X_k}{\times}X_n$, and is characterized as the following fibred product
\[
\begin{tikzcd}
C_n\underset{C_k}{\times}C_n \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &C_n \ar[d,"\trgt_k"]\\
C_n \ar[r,"\src_k"] & C_k.
X_n\underset{X_k}{\times}X_n \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &X_n \ar[d,"\trgt_k"]\\
X_n \ar[r,"\src_k"] & X_k.
\end{tikzcd}
\]
A \emph{morphism of $\oo$-graphs} $f : X \to Y$ is a sequence $(f_n : X_n \to Y_n)_{n \in mathbb{N}}$ of maps that is compatible with source and targets, i.e. for every $n$-cell $x$ of $X$ with $n >0$, we have
\[
f_{n-1}(\src(x))=\src(f_n(x)) \text{ and } f_{n-1}(\trgt(x))=\trgt(f_n(x)).
\]
For an $n$-cell $x$ of $X$, we often write $f(x)$ instead of $f_n(x)$.
The category of $\oo$-graphs and morphisms of $\oo$-graphs is denoted by $\oo\Grph$.
\end{paragr}
\begin{paragr}\label{paragr:defoomagma}
An \emph{$\oo$-magma} consists of an $\oo$-graph $X$ together with maps
\begin{align*}
1_{(\shortminus)}: X_n &\to X_{n+1}\\
x &\mapsto 1_x
\end{align*}
for every $n\geq 0$, and maps
\begin{align*}
(\shortminus)\comp_k(\shortminus): X_n \underset{X_k}{\times}X_n &\to X_n\\
(x,y)&\mapsto x\comp_k y
\end{align*}
for all $0 \leq k <n$, subject to the following axioms:
\begin{enumerate}[label=(\alph*)]
\item For every $n\geq 0$ and every $n$-cell $x$,
\[
\src(1_x)=\src(1_x)=x.
\]
\item For all $0 \leq k< n$ and all $k$-composable $n$-cells $x$ and $y$,
\[
\src(x\comp_k y) =
\begin{cases}
\src(y) &\text{ when }k=n-1,\\
\src(x)\comp_k \src(y) &\text{ otherwise,}
\end{cases}
\]
and
\[
\trgt(x\comp_k y) =
\begin{cases}
\trgt(x) & \text{ when }k=n-1,\\
\trgt(x)\comp_k \trgt(y) &\text{ otherwise.}
\end{cases}
\]
\end{enumerate}
We will use the same letter to denote an $\oo$\nbd-magma and its underlying $\oo$\nbd-graph.
For an $n$-cell $x$, the $(n+1)$-cell $1_{x}$ is referred to as the \emph{unit on $x$}. More generally, for all $0 \leq k < n$, we define maps $\1^n_{(\shortminus)} : C_k \to C_n$ as
\[
\1^{n}_{(\shortminus)} := \underbrace{1_{(\shortminus)} \circ \dots \circ 1_{(\shortminus)}}_{n-k \text{ times }} : C_k \to C_n.
\]
For a $k$-cell $x$ and $n>k$, the $n$-cell $\1^n_x$ is referred to as the \emph{$n$-dimensional unit on $x$}.% and, for consistency, we also set
For two $k$-composable $n$-cells $x$ and $y$, the $n$-cell $x \comp_k y$ is referred to as the \emph{$k$-composition} of $x$ and $y$.
More generally, we extend the notion of $k$-composition for cells of different dimension in the following way. Let $x$ be an $n$-cell, $y$ be an $m$-cell with $m \neq n$ and $k~<~\min\{m,n\}$. The cells $x$ and $y$ are \emph{$k$-composable} if $\src_k(x)=\trgt_k(y)$, in which case we define the cell $x\comp_k y$ of dimension $\max\{m,n\}$ as
\[
x\comp_k y :=
\begin{cases}
1^n_x \comp_k y &\text{ if } m<n\\
x \comp_k 1^m_y &\text{ if } m>n\\
\end{cases}
\]
We also follow the convention that if $n<m$, then $\comp_n$ has priority over $\comp_m$. This means that
\[
x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n z = x \comp_m (y \comp_n z)
\]
whenever these equations make sense.
A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd-graphs that is compatible with units and composition, i.e. for every $n$-cell $x$, we have
\[
f(1_x)=1_{f(x)},
\]
and for every $k$-composable $n$-cells $x$ and $y$, we have
\[
f(x\comp_k y)= f(x) \comp_k f(y).
\]
The category of $\oo$\nbd-magmas and morphisms of $\oo$-magmas is denoted by $\oo\Mag$.
\end{paragr}
\begin{paragr}\label{paragr:defoocat}
An \emph{$\oo$-category} is an $\oo$-magma $X$ that satisfies the following axioms:
\begin{description}
\item[Units:] for all $k<n$, for every $n$-cell $x$, we have
\[
\1^n_{\trgt_k(x)}\comp_k x =x= x \comp_k\1^n_{\src_k(x)},
\]
\item[Functoriality of units:] for all $k<n$ and for all $k$-composable $n$-cells $x$ and $y$, we have
\[
1_{x\comp_k y}=1_{x}\comp_k 1_{y},
\]
\item[Associativity:] for all $k<n$, for all $n$-cells $x, y$ and $z$ such that $x$ and $y$ are $k$-composable, and $y$ and $z$ are $k$-composable, we have
\[
(x\comp_{k}y)\comp_{k}z=x\comp_k(y\comp_kz),
\]
\item[Exchange rule:] for all $k,l,n \in \mathbb{N}$ with $k<n$ and $l<n$, for all $n$-cells $x,x',y$ and $y'$ such that
\begin{itemize}
\item[-] $x$ and $y$ are $l$-composable, $x'$ and $y'$ are $l$-composable,
\item[-] $x$ and $x'$ are $k$-composable, $y$ and $y'$ are $k$-composable,
\end{itemize}
we have
\[
((x \comp_k x')\comp_l (y \comp_k y'))=((x \comp_l y)\comp_k (x' \comp_l y')).
\]
\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.
\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 every $n\geq 0$, there is a canonical functor
\[
\tau_{\leq n}^s : \oo\Cat \to n\Cat
\]
that simply discards all the cells of dimension strictly higher than $n$. This functor has a left adjoint
\[
\iota : n\Cat \to \oo\Cat,
\]
where for an $n$-category $C$, the $\oo$-category $\iota(C)$ has the same $k$-cells as $C$ for $k\leq n$ and only unit cells in dimension strictly higher than $n$. This functor itself has a left adjoint
\[
\tau_{\leq n }^i : \oo\Cat \to n\Cat,
\]
where for an $\oo$-category $C$, the $n$-category $\tau_{\leq n}^i(C)$ has the same $k$-cells as $C$ for $k < n$ and whose set of $n$-cells is the quotient of $C_n$ under the equivalence relation $\sim$ generated by
\[
x \sim y \text{ if there exists } z \in C_{n+1} \text{ of the form } z : x \to y.
\]
The functor $\tau_{\leq n}^s$ also have a right adjoint
\[
\kappa : n\Cat \to \oo\Cat,
\]
where for an $n$-category $C$, the $\oo$-category $\kappa(C)$ has the same $k$-cells as $C$ for $k \leq n$ and has exactly one $m$-cell $x \to y$ for every pair of parallel $(m-1)$-cells $(x,y)$ with $m>n$.
The sequence of adjunctions
\[
\kappa \dashv \tau^s_{\leq n} \dashv \iota \dashv \tau^{i}_{\leq n}
\]
is maximal in that $\kappa$ doesn't have a left adjoint and $\tau^{i}_{\leq n}$ doesn't have right adjoint.
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.
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 $C$ there is a canonical filtration
\[
\sk_{0}(C) \to \sk_{1}(C) \to \cdots \to \sk_{n}(C) \to \cdots,
\]
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}
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