1073 \label{lem:colim-injective} |
1073 \label{lem:colim-injective} |
1074 Let $W$ be a manifold of dimension less than $n$. Then for each |
1074 Let $W$ be a manifold of dimension less than $n$. Then for each |
1075 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
1075 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
1076 \end{lem} |
1076 \end{lem} |
1077 \begin{proof} |
1077 \begin{proof} |
1078 \nn{...} |
1078 $\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
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1079 injective. |
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1080 Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
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1081 modulo the relation which identifies the domain of each of the injective maps |
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1082 with it's image. |
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1083 |
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1084 To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
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1085 |
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1086 Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$. |
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1087 Then there exist |
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1088 \begin{itemize} |
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1089 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
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1090 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
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1091 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
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1092 such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$. |
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1093 \end{itemize} |
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1094 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
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1095 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
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1096 disjoint union of balls, and then invoke the associativity axiom \ref{nca-assoc}. |
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1097 \nn{hmmm... it would be nicer if this were ``7.xx" instead of ``4"} |
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1098 |
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1099 Let $z$ be a decomposition of $W$ which is in general position with respect to all of the |
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1100 $x_i$'s and $v_i$'s. |
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1101 There there decompositions $x'_i$ and $v'_i$ (for all $i$) such that |
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1102 \begin{itemize} |
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1103 \item $x'_i$ antirefines to $x_i$ and $z$; |
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1104 \item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$; |
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1105 \item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and |
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1106 \item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image |
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1107 of $b'_i$ and $b'_{i+1}$. |
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1108 \end{itemize} |
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1109 Now consider the diagrams |
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1110 \[ \xymatrix{ |
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1111 & \psi(x'_{i-1}) \ar[rd] & \\ |
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1112 \psi(v'_i) \ar[ru] \ar[rd] & & \psi(z) \\ |
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1113 & \psi(x'_i) \ar[ru] & |
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1114 } \] |
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1115 The associativity axiom applied to this diagram implies that $a'_{i-1}$ and $a'_i$ |
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1116 map to the same element $c\in \psi(z)$. |
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1117 Therefore $a'_0$ and $a'_k$ both map to $c$. |
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1118 But $a'_0$ and $a'_k$ are both elements of $\psi(x'_0)$ (because $x'_k = x'_0$). |
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1119 So by the injectivity clause of the composition axiom, we must have that $a'_0 = a'_k$. |
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1120 But this implies that $a = a_0 = a_k = \hat{a}$, contrary to our assumption that $a\ne \hat{a}$. |
1079 \end{proof} |
1121 \end{proof} |
1080 |
1122 |
1081 \nn{need to finish explaining why we have a system of fields; |
1123 \nn{need to finish explaining why we have a system of fields; |
1082 define $k$-cat $\cC(\cdot\times W)$} |
1124 define $k$-cat $\cC(\cdot\times W)$} |
1083 |
1125 |