1035 with the gluing maps $M_i\to M_{i+1}$ and $M'_i\to M'_{i+1}$. |
1035 with the gluing maps $M_i\to M_{i+1}$ and $M'_i\to M'_{i+1}$. |
1036 |
1036 |
1037 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1037 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1038 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1038 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1039 with $\du_b Y_b = M_i$ for some $i$, |
1039 with $\du_b Y_b = M_i$ for some $i$, |
1040 and with $M_0,\ldots, M_i$ each being a disjoint union of balls. |
1040 and with $M_0, M_1, \ldots, M_i$ each being a disjoint union of balls. |
1041 |
1041 |
1042 \begin{defn} |
1042 \begin{defn} |
1043 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
1043 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
1044 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
1044 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
1045 See Figure \ref{partofJfig} for an example. |
1045 See Figure \ref{partofJfig} for an example. |
1054 \end{figure} |
1054 \end{figure} |
1055 |
1055 |
1056 An $n$-category $\cC$ determines |
1056 An $n$-category $\cC$ determines |
1057 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
1057 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
1058 (possibly with additional structure if $k=n$). |
1058 (possibly with additional structure if $k=n$). |
1059 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
1059 For pedagogical reasons, let us first the case where a decomposition $y$ of $W$ is a nice, non-pathological |
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1060 cell decomposition. |
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1061 Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls, |
1060 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
1062 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
1061 are splittable along this decomposition. |
1063 are splittable along this decomposition. |
1062 |
1064 |
1063 \begin{defn} |
1065 We can now |
1064 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
1066 define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
1065 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
1067 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
1066 \begin{equation} |
1068 \begin{equation} |
1067 \label{eq:psi-C} |
1069 %\label{eq:psi-C} |
1068 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
1070 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
1069 \end{equation} |
1071 \end{equation} |
1070 where the restrictions to the various pieces of shared boundaries amongst the cells |
1072 where the restrictions to the various pieces of shared boundaries amongst the cells |
1071 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells). |
1073 $X_a$ 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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1074 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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1075 |
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1076 In general, $y$ might be more general than a cell decomposition |
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1077 (see Example \ref{sin1x-example}), so we must define $\psi_{\cC;W}$ in a more roundabout way. |
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1078 \nn{...} |
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1079 |
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1080 \begin{defn} |
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1081 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
|
1082 \nn{...} |
1072 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 $\cC$. |
1083 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 $\cC$. |
1073 \end{defn} |
1084 \end{defn} |
1074 |
1085 |
1075 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1086 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1076 we need to say a bit more. |
1087 we need to say a bit more. |