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 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
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1071 Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection. |
1071 |
1072 |
1072 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1073 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1073 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1074 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1074 We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1075 We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1075 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1076 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1086 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1087 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1087 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1088 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1088 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1089 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1089 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1090 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1090 The $i$-th condition is defined similarly. |
1091 The $i$-th condition is defined similarly. |
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1092 Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$. |
1091 |
1093 |
1092 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1094 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1093 above conditions for all $i$ and also all |
1095 above conditions for all $i$ and also all |
1094 ball decompositions compatible with $x$. |
1096 ball decompositions compatible with $x$. |
1095 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
1097 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
1096 compatibility for one ball decomposition implies gluing compatibility for all other ball decompositions. |
1098 compatibility for one ball decomposition implies gluing compatibility for all other ball decompositions. |
1097 Rather than try to prove a similar result for arbitrary |
1099 Rather than try to prove a similar result for arbitrary |
1098 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
1100 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
1099 |
1101 |
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 $\cC$. |
1102 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ |
1101 |
1103 is given by the composition maps of $\cC$. |
1102 |
1104 This completes the definition of the functor $\psi_{\cC;W}$. |
1103 \nn{to do: define splittability and restrictions for colimits} |
1105 |
1104 |
1106 Note that we have constructed, at the last stage of the above procedure, |
1105 \noop{ %%%%%%%%%%%%%%%%%%%%%%% |
1107 a map from $\psi_{\cC;W}(x)$ to $\cl\cC(\bd M_m) = \cl\cC(\bd W)$. |
1106 For pedagogical reasons, let us first consider the case of a decomposition $y$ of $W$ |
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1107 which is a nice, non-pathological cell decomposition. |
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1108 Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls, |
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1109 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
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1110 are splittable along this decomposition. |
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1111 |
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1112 We can now |
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1113 define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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1114 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
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1115 \begin{equation} |
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1116 %\label{eq:psi-C} |
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1117 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
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1118 \end{equation} |
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1119 where the restrictions to the various pieces of shared boundaries amongst the cells |
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1120 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells). |
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1121 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 $\cC$. |
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1122 |
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1123 In general, $y$ might be more general than a cell decomposition |
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1124 (see Example \ref{sin1x-example}), so we must define $\psi_{\cC;W}$ in a more roundabout way. |
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1125 \nn{...} |
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1126 |
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1127 \begin{defn} |
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1128 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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1129 \nn{...} |
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1130 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 $\cC$. |
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1131 \end{defn} |
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1132 } % end \noop %%%%%%%%%%%%%%%%%%%%%%% |
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1133 |
1108 |
1134 |
1109 |
1135 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1110 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1136 we need to say a bit more. |
1111 we need to say a bit more. |
1137 We can rewrite the colimit as |
1112 We can rewrite the colimit as |
1138 \begin{equation} \label{eq:psi-CC} |
1113 \[ % \begin{equation} \label{eq:psi-CC} |
1139 \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
1114 \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
1140 \end{equation} |
1115 \] % \end{equation} |
1141 where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
1116 where $\beta$ runs through |
1142 (which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |
1117 boundary conditions on $\du_a X_a$ which are compatible with gluing as specified above |
1143 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
1118 and $\cC(X_a; \beta)$ |
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1119 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agrees with $\beta$. |
1144 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
1120 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
1145 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
1121 $\cS$ and the coproduct and product in the above expression should be replaced by the appropriate |
1146 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
1122 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
1147 |
1123 |
1148 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1124 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1149 |
1125 |
1150 \begin{defn}[System of fields functor] |
1126 \begin{defn}[System of fields functor] |