recyclebin.tex 19.9 KB
 Leonard Guetta committed Jan 10, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168  \begin{paragr}\label{paragr:defoograph} An \emph{$\oo$-graph} $C$ consists of a sequence $(C_k)_{k \in \mathbb{N}}$ of sets together with maps $\begin{tikzcd} C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k} \end{tikzcd}$ for every $k > 0$, subject to the \emph{globular identities}: \begin{equation*} \left\{ \begin{aligned} s \circ s &= s \circ t, \\ t \circ t &= t \circ s. \end{aligned} \right. \end{equation*} Elements of $C_k$ are called \emph{$k$-cells} or \emph{cells of dimension $k$}. For $x$ a $k$-cell with $k>0$, $s(x)$ is the \emph{source} of $x$ and $t(x)$ is the \emph{target} of $x$. More generally for all $k0$ and $s(x)=s(y) \text{ and } t(x)=t(y).$ For all $k0$, the squares $\begin{tikzcd} C_k \ar[d,"s"] \ar[r,"f_k"]&C'_k \ar[d,"s"] \\ C_{k-1} \ar[r,"f_{k-1}"] & C'_{k-1} \end{tikzcd} \quad \begin{tikzcd} C_k \ar[d,"t"] \ar[r,"f_k"]&C'_k \ar[d,"t"] \\ C_{k-1} \ar[r,"f_{k-1}"] & C'_{k-1} \end{tikzcd}$ are commutative. For a $k$-cell $x$, we will often write $f(x)$ instead of $f_k(x)$. We denote by $\ooGrph$ the category of $\oo$-graphs and morphisms of $\oo$-graphs. \end{paragr} \begin{paragr} For $n \in \mathbb{N}$, the notion of \emph{$n$-graph} is defined similarly, only this time there is only a finite sequence $(C_k)_{0 \leq k \leq n}$ of cells. For example, a $0$-graph is just a set and a $1$-graph is an ordinary graph $\begin{tikzcd} C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_1. \end{tikzcd}$ The definition of morphism of $n$-graphs is the same as for $\omega$-graphs, only this time there is only a finite sequence $(f_k : C_k \to C'_k)_{0 \leq k \leq n}$ of maps. We denote by $\nGrph$ the category of $n$-graphs and morphisms of $n$-graphs. \end{paragr} \begin{paragr} Let $n \in \nbar$. An \emph{$n$-magma} consists of: \begin{itemize} \item[-] an $n$-graph $C$, \item[-] maps \begin{aligned} (\shortminus)\underset{k}{\ast}(\shortminus) : C_l\underset{C_k}{\times}C_l &\to C_l \\ (x,y) &\mapsto x\underset{k}{\ast}y \end{aligned} for all $l,k \in \mathbb{N}$ with $k < l \leq n$,\footnote{Note that if $n=\omega$, then $l0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product \label{squarecellext} \begin{tikzcd} n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\ n\Grph \ar[r,"\tau"] & (n \shortminus 1)\Grph, \end{tikzcd} where the right vertical arrow is the obvious forgetful functor. More concretely, an $n$-cellular extension can be encoded in the data of a quadruple $(\Sigma,C,s,t)$ where $\Sigma$ is a set, $C$ is a $(n\shortminus 1)$-category, $s$ and $t$ are maps $s,t : \Sigma \to C_n$ such that $\begin{tikzcd} C_{n-1} & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] C_n & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] \Sigma \end{tikzcd}$ satisfy the globular identities. Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with extra $n$-cells that make it a $n$-graph. We will sometimes write $\begin{tikzcd} C &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] \Sigma \end{tikzcd}$ to denote an $n$-cellular extension $(\Sigma,C,s,t)$. \remtt{Est-ce que je garde cette notation ?} A morphism of $n$-cellular extensions from $(\Sigma,C,s,t)$ to $(\Sigma',C',s',t')$ consists of a pair $(\varphi,f)$ where $f : C \to C'$ is a morphism of $(n\shortminus 1)\Cat$ and $\varphi : \Sigma \to \Sigma'$ is a map such that the squares $\begin{tikzcd} \Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\ C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1} \end{tikzcd} \text{ and } \begin{tikzcd} \Sigma \ar[r,"\varphi"] \ar[d,"t"] & \Sigma' \ar[d,"t'"] \\ C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1} \end{tikzcd}$ commute. Once again, we will use the notation $\tau$ for the functor \begin{aligned} \tau : n\CellExt &\to (n\shortminus 1)\Cat\\ (\Sigma,C,s,t) &\mapsto C \end{aligned} which is simply the top horizontal arrow of square \eqref{squarecellext}. \end{paragr} \begin{paragr} Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product \label{squareprecat} \begin{tikzcd} n\PCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\ n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag. \end{tikzcd} More concretely, objects of $n\PCat$ can be seen as $n$-magmas such that if we forget their $n$-cells then they satisfy the axioms of $(n\shortminus 1)$-category. The left vertical arrow of the previous square is easily seen to be full, and we will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$. The top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation $\tau : n\PCat \to (n\shortminus 1)\Cat.$ The commutative square $\begin{tikzcd} n\Cat\ar[r,"\tau"] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\ n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag, \end{tikzcd}$ where the vertical arrows are the obvious forgetful functors, induces a canonical functor $V : n\Cat \to n\PCat,$ which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$. Moreover, the canonical commutative diagram $\begin{tikzcd} n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\ n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l] \end{tikzcd}$ induces a canonical functor $W : n\PCat \to n\CellExt.$ For an $n$-precategory $C$, $W(C)$ is simply the cellular extension $\begin{tikzcd} \tau(C) &\ar[l,shift right,"s"'] \ar[l,shift left,"t"] C_n. \end{tikzcd}$ Finally, we define the functor $U := W \circ V : n\Cat \to n\CellExt.$ The relation between $n\Cat$, $n\PCat$, $n\CellExt$ and $(n\shortminus 1)\Cat$ is summed up in the following commutative diagram: $\begin{tikzcd} n\Cat \ar[rr,bend left,"U"]\ar[r,"V"] \ar[rrd,"\tau"'] & n\PCat \ar[r,"W"]\ar[rd,"\tau"] & n\CellExt \ar[d,"\tau"] \\ &&(n\shortminus 1)\Cat. \end{tikzcd}$ We will now explicitely construct a left adjoint of $U$. In order to do that, we will successively construct left adjoints of $W$ and $V$. \end{paragr} \begin{paragr} Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with \begin{itemize} \item[-] $\tau(W_!(E))=C$, \item[-] $W_!(E)_n=\Sigma^{+}$, \item[-] source and target maps $\Sigma^+ \to C_{n-1}$ as defined in the previous paragraph, \item[-] for every $x \in C_{n-1}$, $1_x := (\ii_x)$ \item[-] for every $v,w \in \Sigma^+$ that are $k$-composable for a \$k