1032 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1032 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1033 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1033 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1034 |
1034 |
1035 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
1035 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
1036 is more involved. |
1036 is more involved. |
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1037 We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$. |
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1038 The first is the usual one, which works for any indexing category. |
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1039 The second construction, we we call the {\it local} homotopy colimit, |
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1040 \nn{give it a different name?} |
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1041 is more closely related to the blob complex |
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1042 construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties |
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1043 of the indexing category $\cell(W)$. |
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1044 |
1037 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
1045 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
1038 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1046 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1039 Define $\cl{\cC}(W)$ as a vector space via |
1047 Define $\cl{\cC}(W)$ as a vector space via |
1040 \[ |
1048 \[ |
1041 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1049 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1049 \[ |
1057 \[ |
1050 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1058 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1051 \] |
1059 \] |
1052 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1060 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1053 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1061 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1054 %\nn{need to say this better} |
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1055 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1062 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1056 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1063 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1057 %of $A_\infty$ category} |
1064 %of $A_\infty$ category} |
1058 |
1065 |
1059 We can think of this construction as starting with a disjoint copy of a complex for each |
1066 We can think of this construction as starting with a disjoint copy of a complex for each |
1060 permissible decomposition (the 0-simplices). |
1067 permissible decomposition (the 0-simplices). |
1061 Then we glue these together with mapping cylinders coming from gluing maps |
1068 Then we glue these together with mapping cylinders coming from gluing maps |
1062 (the 1-simplices). |
1069 (the 1-simplices). |
1063 Then we kill the extra homology we just introduced with mapping |
1070 Then we kill the extra homology we just introduced with mapping |
1064 cylinders between the mapping cylinders (the 2-simplices), and so on. |
1071 cylinders between the mapping cylinders (the 2-simplices), and so on. |
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1072 |
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1073 Next we describe the local homotopy colimit. |
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1074 This is similar to the usual homotopy colimit, but using |
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1075 a cone-product set (Remark \ref{blobsset-remark}) in place of a simplicial set. |
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1076 The cone-product $m$-polyhedra for the set are pairs $(x, E)$, where $x$ is a decomposition of $W$ |
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1077 and $E$ is an $m$-blob diagram such that each blob is a union of balls of $x$. |
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1078 (Recall that this means that the interiors of |
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1079 each pair of blobs (i.e.\ balls) of $E$ are either disjoint or nested.) |
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1080 To each $(x, E)$ we associate the chain complex $\psi_{\cC;W}(x)$, shifted in degree by $m$. |
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1081 The boundary has a term for omitting each blob of $E$. |
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1082 If we omit an innermost blob then we replace $x$ by the formal difference $x - \gl(x)$, where |
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1083 $\gl(x)$ is obtained from $x$ by gluing together the balls of $x$ contained in the blob we are omitting. |
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1084 The gluing maps of $\cC$ give us a maps from $\psi_{\cC;W}(x)$ to $\psi_{\cC;W}(\gl(x))$. |
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1085 |
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1086 One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
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1087 Eilenberg-Zilber type subdivision argument. |
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1088 |
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1089 \medskip |
1065 |
1090 |
1066 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1091 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1067 |
1092 |
1068 It is easy to see that |
1093 It is easy to see that |
1069 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
1094 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |