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 |
999 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
999 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1000 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1000 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1001 |
1001 |
1002 \medskip |
1002 \medskip |
1003 |
1003 |
1004 We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1004 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1005 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1005 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1006 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1006 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1007 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1007 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1008 We'll later give a more explicit description of this colimit. |
1008 We'll later give a more explicit description of this colimit. |
1009 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1009 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
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 For pedagogical reasons, let us first the case where a decomposition $y$ of $W$ is a nice, non-pathological |
1059 Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$). |
1060 cell decomposition. |
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 $\bd X_b$. |
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1063 (Keep in mind that perhaps $a=b$.) |
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1064 Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy |
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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 $\prod_a \cC(P_a)$.) |
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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 \[ |
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1077 \prod_a \cC(X_a) \to \cl\cC(\bd M_0) . |
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1078 \] |
|
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 |
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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 |
|
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 |
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1101 |
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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. |
1061 Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls, |
1109 Then each $k$-ball $X$ of $y$ has its boundary decomposed into $k{-}1$-balls, |
1062 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 |
1063 are splittable along this decomposition. |
1111 are splittable along this decomposition. |
1064 |
1112 |
1065 We can now |
1113 We can now |
1080 \begin{defn} |
1128 \begin{defn} |
1081 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
1129 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
1082 \nn{...} |
1130 \nn{...} |
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 $\cC$. |
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 $\cC$. |
1084 \end{defn} |
1132 \end{defn} |
|
1133 } % end \noop %%%%%%%%%%%%%%%%%%%%%%% |
|
1134 |
1085 |
1135 |
1086 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, |
1087 we need to say a bit more. |
1137 we need to say a bit more. |
1088 We can rewrite Equation \ref{eq:psi-C} as |
1138 We can rewrite the colimit as |
1089 \begin{equation} \label{eq:psi-CC} |
1139 \begin{equation} \label{eq:psi-CC} |
1090 \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) , |
1091 \end{equation} |
1141 \end{equation} |
1092 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 |
1093 (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 $\cC(X_a; \beta)$ |