671 \end{equation*} |
671 \end{equation*} |
672 where $K$ is the vector space spanned by elements $a - g(a)$, with |
672 where $K$ is the vector space spanned by elements $a - g(a)$, with |
673 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
673 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
674 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
674 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
675 |
675 |
676 \nn{ ** resume revising here} |
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677 |
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678 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
676 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
679 is more involved. |
677 is more involved. |
680 %\nn{should probably rewrite this to be compatible with some standard reference} |
678 %\nn{should probably rewrite this to be compatible with some standard reference} |
681 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
679 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
682 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
680 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
683 Define $V$ as a vector space via |
681 Define $V$ as a vector space via |
684 \[ |
682 \[ |
685 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
683 V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
686 \] |
684 \] |
687 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are obtuse: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
685 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) |
688 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
686 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
689 summands plus another term using the differential of the simplicial set of $m$-sequences. |
687 summands plus another term using the differential of the simplicial set of $m$-sequences. |
690 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
688 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
691 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
689 summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
692 \[ |
690 \[ |
693 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
691 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
694 \] |
692 \] |
695 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
693 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
696 is the usual gluing map coming from the antirefinement $x_0 < x_1$. |
694 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
697 \nn{need to say this better} |
695 \nn{need to say this better} |
698 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
696 \nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
699 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
697 combine only two balls at a time; for $n=1$ this version will lead to usual definition |
700 of $A_\infty$ category} |
698 of $A_\infty$ category} |
701 |
699 |
702 We will call $m$ the filtration degree of the complex. |
700 We will call $m$ the filtration degree of the complex. |
703 We can think of this construction as starting with a disjoint copy of a complex for each |
701 We can think of this construction as starting with a disjoint copy of a complex for each |
704 permissible decomposition (filtration degree 0). |
702 permissible decomposition (filtration degree 0). |
705 Then we glue these together with mapping cylinders coming from gluing maps |
703 Then we glue these together with mapping cylinders coming from gluing maps |
706 (filtration degree 1). |
704 (filtration degree 1). |
707 Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2). |
705 Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2). |
708 And so on. |
706 And so on. |
709 |
707 |
710 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
708 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
711 |
709 |
712 It is easy to see that |
710 It is easy to see that |
724 |
722 |
725 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
723 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
726 a.k.a.\ actions). |
724 a.k.a.\ actions). |
727 The definition will be very similar to that of $n$-categories. |
725 The definition will be very similar to that of $n$-categories. |
728 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
726 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
729 \nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
727 %\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
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728 |
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729 \nn{** resume revising here} |
730 |
730 |
731 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
731 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
732 in the context of an $m{+}1$-dimensional TQFT. |
732 in the context of an $m{+}1$-dimensional TQFT. |
733 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
733 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
734 This will be explained in more detail as we present the axioms. |
734 This will be explained in more detail as we present the axioms. |