1065 (see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. |
1065 (see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. |
1066 |
1066 |
1067 Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$. |
1067 Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$. |
1068 (To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is |
1068 (To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is |
1069 a 0-ball, to be $\prod_a \cC(P_a)$.) |
1069 a 0-ball, to be $\prod_a \cC(P_a)$.) |
|
1070 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
1070 |
1071 |
1071 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1072 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1072 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1073 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1073 We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1074 We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1074 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1075 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1096 Rather than try to prove a similar result for arbitrary |
1097 Rather than try to prove a similar result for arbitrary |
1097 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
1098 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
1098 |
1099 |
1099 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
1100 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
1100 |
1101 |
1101 |
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1102 \nn{...} |
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1103 |
1102 |
1104 \nn{to do: define splittability and restrictions for colimits} |
1103 \nn{to do: define splittability and restrictions for colimits} |
1105 |
1104 |
1106 \noop{ %%%%%%%%%%%%%%%%%%%%%%% |
1105 \noop{ %%%%%%%%%%%%%%%%%%%%%%% |
1107 For pedagogical reasons, let us first consider the case of a decomposition $y$ of $W$ |
1106 For pedagogical reasons, let us first consider the case of a decomposition $y$ of $W$ |