### 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 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 $0\leq k 0$, 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 $0 \leq k k$, 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 $x\comp_k y := \begin{cases} 1^n_x \comp_k y &\text{ if } mn\\ \end{cases}$ We also follow the convention that if $nn$. 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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