436 to $\bd X$. |
436 to $\bd X$. |
437 For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
437 For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
438 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
438 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
439 $W\to W'$ which restricts to the identity on the boundary. |
439 $W\to W'$ which restricts to the identity on the boundary. |
440 |
440 |
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441 \item \nn{sphere modules; ref to below} |
441 |
442 |
442 \end{itemize} |
443 \end{itemize} |
443 |
444 |
444 |
445 |
445 Examples of $A_\infty$ $n$-categories: |
446 Examples of $A_\infty$ $n$-categories: |
827 |
828 |
828 \subsection{Modules as boundary labels} |
829 \subsection{Modules as boundary labels} |
829 \label{moddecss} |
830 \label{moddecss} |
830 |
831 |
831 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
832 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
832 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
833 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
833 component $\bd_i W$ of $W$. |
834 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
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835 |
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836 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
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837 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
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838 %component $\bd_i W$ of $W$. |
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839 %(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
834 |
840 |
835 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
841 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
836 \nn{give ref} |
842 \nn{give ref} |
837 (If $k = n$ and our $k$-categories are enriched, then |
843 (If $k = n$ and our $k$-categories are enriched, then |
838 $\cC(W, \cN)$ will have additional structure; see below.) |
844 $\cC(W, \cN)$ will have additional structure; see below.) |
841 \[ |
847 \[ |
842 W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
848 W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
843 \] |
849 \] |
844 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
850 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
845 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
851 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
846 with $M_{ib}\cap\bd_i W$ being the marking. |
852 with $M_{ib}\cap Y_i$ being the marking. |
847 \nn{need figure} |
853 (See Figure \ref{mblabel}.) |
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854 \begin{figure}[!ht]\begin{equation*} |
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855 \mathfig{.9}{tempkw/mblabel} |
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856 \end{equation*}\caption{A permissible decomposition of a manifold |
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857 whose boundary components are labeled my $\cC$ modules $\{\cN_i\}$.}\label{mblabel}\end{figure} |
848 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
858 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
849 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
859 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
850 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
860 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
851 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
861 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
852 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
862 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
863 (Think fibered product.) |
873 (Think fibered product.) |
864 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
874 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
865 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
875 via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
866 |
876 |
867 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
877 Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$. |
868 In other words, for each decomposition $x$ there is a map |
878 (Recall that if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
869 $\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps |
879 homotopy colimit.) |
870 above, and $\cC(W, \cN)$ is universal with respect to these properties. |
880 |
871 |
881 If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
872 More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$. |
882 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
873 \nn{need to say more?} |
883 $D\times Y_i \sub \bd(D\times W)$. |
874 |
884 |
875 \nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.} |
885 It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
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886 has the structure of an $n{-}k$-category. |
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887 We will use a simple special case of this construction in the next subsection to define tensor products |
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888 of modules. |
876 |
889 |
877 \subsection{Tensor products} |
890 \subsection{Tensor products} |
878 |
891 |
879 Next we consider tensor products. |
892 Next we consider tensor products. |
880 |
893 |