934 (We require that the interiors of the little balls be disjoint, but their |
934 (We require that the interiors of the little balls be disjoint, but their |
935 boundaries are allowed to meet. |
935 boundaries are allowed to meet. |
936 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
936 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
937 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
937 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
938 (But note also that this inclusion is not |
938 (But note also that this inclusion is not |
939 necessarily a homotopy equivalence.) |
939 necessarily a homotopy equivalence.)) |
940 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
940 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
941 by shrinking the little balls (precomposing them with dilations), |
941 by shrinking the little balls (precomposing them with dilations), |
942 we see that both operads are homotopic to the space of $k$ framed points |
942 we see that both operads are homotopic to the space of $k$ framed points |
943 in $B^n$. |
943 in $B^n$. |
944 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
944 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
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}. |
1046 \end{defn} |
1046 \end{defn} |
1047 |
1047 |
1048 \begin{figure}[t] |
1048 \begin{figure}[t] |
1049 \begin{equation*} |
1049 \begin{equation*} |
1050 \mathfig{.63}{ncat/zz2} |
1050 \mathfig{.63}{ncat/zz2} |
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 Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$). |
|
1060 We will define $\psi_{\cC;W}(x)$ to be a certain subset of $\prod_a \cC(X_a)$. |
|
1061 Roughly speaking, $\psi_{\cC;W}(x)$ is the subset where the restriction maps from |
|
1062 $\cC(X_a)$ and $\cC(X_b)$ agree whenever some part of $\bd X_a$ is glued to some part of $\bd X_b$. |
|
1063 (Keep in mind that perhaps $a=b$.) |
|
1064 Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy |
|
1065 (see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. |
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1066 |
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1067 Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$. |
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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 |
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1069 a 0-ball, to be $\prod_a \cC(P_a)$.) |
|
1070 |
|
1071 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
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1072 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
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1073 We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
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1074 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
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1075 By Axiom \ref{nca-boundary}, we have a map |
|
1076 \[ |
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1077 \prod_a \cC(X_a) \to \cl\cC(\bd M_0) . |
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1078 \] |
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1079 The first condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_0)$ is splittable |
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1080 along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\cl\cC(Y_0)$ and $\cl\cC(Y'_0)$ agree |
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1081 (with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map). |
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1082 |
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1083 On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction |
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1084 map to $\cl\cC(N_0)$ which we can compose with the gluing map |
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1085 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
|
1086 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
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1087 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
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1088 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
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1089 The $i$-th condition is defined similarly. |
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1090 |
|
1091 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
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1092 above conditions for all $i$ and also all |
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1093 ball decompositions compatible with $x$. |
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1094 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
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1095 compatibility for one ball decomposition implies gluing compatibility for all other ball decompositions. |
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1096 Rather than try to prove a similar result for arbitrary |
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1097 permissible decompositions, we instead require compatibility with all ways of gluing up the decomposition.) |
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1098 |
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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 $\cC$. |
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1100 |
|
1101 |
|
1102 \nn{...} |
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1103 |
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1104 \nn{to do: define splittability and restrictions for colimits} |
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1105 |
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1106 \noop{ %%%%%%%%%%%%%%%%%%%%%%% |
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1107 For pedagogical reasons, let us first consider the case of a decomposition $y$ of $W$ |
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1108 which is a nice, non-pathological cell decomposition. |
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1109 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 |
1110 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
1061 are splittable along this decomposition. |
1111 are splittable along this decomposition. |
1062 |
1112 |
1063 \begin{defn} |
1113 We can now |
1064 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
1114 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 |
1115 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
1066 \begin{equation} |
1116 \begin{equation} |
1067 \label{eq:psi-C} |
1117 %\label{eq:psi-C} |
1068 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
1118 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
1069 \end{equation} |
1119 \end{equation} |
1070 where the restrictions to the various pieces of shared boundaries amongst the cells |
1120 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). |
1121 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells). |
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 $\cC$. |
1122 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 $\cC$. |
|
1123 |
|
1124 In general, $y$ might be more general than a cell decomposition |
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1125 (see Example \ref{sin1x-example}), so we must define $\psi_{\cC;W}$ in a more roundabout way. |
|
1126 \nn{...} |
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1127 |
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1128 \begin{defn} |
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1129 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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1130 \nn{...} |
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1131 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} |
1132 \end{defn} |
|
1133 } % end \noop %%%%%%%%%%%%%%%%%%%%%%% |
|
1134 |
1074 |
1135 |
1075 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1136 If $k=n$ in the above definition and we are enriching in some auxiliary category, |
1076 we need to say a bit more. |
1137 we need to say a bit more. |
1077 We can rewrite Equation \ref{eq:psi-C} as |
1138 We can rewrite the colimit as |
1078 \begin{equation} \label{eq:psi-CC} |
1139 \begin{equation} \label{eq:psi-CC} |
1079 \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
1140 \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
1080 \end{equation} |
1141 \end{equation} |
1081 where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
1142 where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
1082 (which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |
1143 (which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |