991 \end{defn} |
991 \end{defn} |
992 |
992 |
993 We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
993 We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
994 with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
994 with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
995 |
995 |
996 We now give a more concrete description of the colimit in each case. |
996 We now give more concrete descriptions of the above colimits. |
997 If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, |
997 |
998 we can take the vector space $\cl{\cC}(W,c)$ to be the direct sum over all permissible decompositions of $W$ |
998 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
|
999 the colimit is |
|
1000 \[ |
|
1001 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim , |
|
1002 \] |
|
1003 where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
|
1004 induced by refinement and gluing. |
|
1005 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
|
1006 we can take |
999 \begin{equation*} |
1007 \begin{equation*} |
1000 \cl{\cC}(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K |
1008 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K |
1001 \end{equation*} |
1009 \end{equation*} |
1002 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1010 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1003 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1011 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1004 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1012 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1005 |
1013 |
1013 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1021 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1014 \] |
1022 \] |
1015 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1023 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1016 (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
1024 (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
1017 the complex $U[m]$ is concentrated in degree $m$.) |
1025 the complex $U[m]$ is concentrated in degree $m$.) |
|
1026 \nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?} |
1018 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1027 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1019 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1028 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1020 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1029 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1021 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1030 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1022 \[ |
1031 \[ |
1023 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1032 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1024 \] |
1033 \] |
1025 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1034 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1026 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1035 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1027 \nn{need to say this better} |
1036 %\nn{need to say this better} |
1028 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1037 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1029 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1038 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1030 of $A_\infty$ category} |
1039 %of $A_\infty$ category} |
1031 |
1040 |
1032 We will call $m$ the filtration degree of the complex. |
1041 We will call $m$ the filtration degree of the complex. |
|
1042 \nn{is there a more standard term for this?} |
1033 We can think of this construction as starting with a disjoint copy of a complex for each |
1043 We can think of this construction as starting with a disjoint copy of a complex for each |
1034 permissible decomposition (filtration degree 0). |
1044 permissible decomposition (filtration degree 0). |
1035 Then we glue these together with mapping cylinders coming from gluing maps |
1045 Then we glue these together with mapping cylinders coming from gluing maps |
1036 (filtration degree 1). |
1046 (filtration degree 1). |
1037 Then we kill the extra homology we just introduced with mapping |
1047 Then we kill the extra homology we just introduced with mapping |
1038 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1048 cylinders between the mapping cylinders (filtration degree 2), and so on. |
1039 |
1049 |
1040 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1050 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1041 |
1051 |
1042 It is easy to see that |
1052 It is easy to see that |
1043 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
1053 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
1044 comprise a natural transformation of functors. |
1054 comprise a natural transformation of functors. |
1045 |
1055 |
1046 \begin{lem} |
1056 \begin{lem} |
1047 \label{lem:colim-injective} |
1057 \label{lem:colim-injective} |
1048 Let $W$ be a manifold of dimension less than $n$. Then for each |
1058 Let $W$ be a manifold of dimension less than $n$. Then for each |