584 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
584 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
585 and $C_*$ denotes singular chains. |
585 and $C_*$ denotes singular chains. |
586 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
586 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
587 \end{example} |
587 \end{example} |
588 |
588 |
589 See \ref{thm:map-recon} below, recovering $C_*(\Maps{M \to T})$ as (up to homotopy) the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
589 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps{M \to T})$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
590 |
590 |
591 \begin{example}[Blob complexes of balls (with a fiber)] |
591 \begin{example}[Blob complexes of balls (with a fiber)] |
592 \rm |
592 \rm |
593 \label{ex:blob-complexes-of-balls} |
593 \label{ex:blob-complexes-of-balls} |
594 Fix an $m$-dimensional manifold $F$. |
594 Fix an $m$-dimensional manifold $F$. We will define an $A_\infty$ $n-m$-category $\cC$. |
595 Given a plain $n$-category $C$, |
595 Given a plain $n$-category $C$, |
596 when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
596 when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
597 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
597 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
598 where $\bc^C_*$ denotes the blob complex based on $C$. |
598 where $\bc^C_*$ denotes the blob complex based on $C$. |
599 \end{example} |
599 \end{example} |
600 |
600 |
601 This example will be essential for ???, which relates ... |
601 This example will be essential for Theorem \ref{product_thm} below, which relates ... |
602 |
602 |
603 \begin{example} |
603 \begin{example} |
604 \nn{should add $\infty$ version of bordism $n$-cat} |
604 \nn{should add $\infty$ version of bordism $n$-cat} |
605 \end{example} |
605 \end{example} |
606 |
606 |
778 (The union is along $N\times \bd W$.) |
778 (The union is along $N\times \bd W$.) |
779 (If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
779 (If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
780 the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
780 the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
781 |
781 |
782 \begin{figure}[!ht] |
782 \begin{figure}[!ht] |
783 $$\mathfig{.8}{tempkw/blah15}$$ |
783 $$\mathfig{.8}{ncat/boundary-collar}$$ |
784 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
784 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
785 |
785 |
786 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
786 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
787 Call such a thing a {marked $k{-}1$-hemisphere}. |
787 Call such a thing a {marked $k{-}1$-hemisphere}. |
788 |
788 |
991 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
991 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
992 there is no left/right module distinction. |
992 there is no left/right module distinction. |
993 |
993 |
994 \medskip |
994 \medskip |
995 |
995 |
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996 We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
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997 |
996 Examples of modules: |
998 Examples of modules: |
997 \begin{itemize} |
999 \begin{itemize} |
998 \item \nn{examples from TQFTs} |
1000 \item \nn{examples from TQFTs} |
999 \item \nn{for maps to $T$, can restrict to subspaces of $T$;} |
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1000 \end{itemize} |
1001 \end{itemize} |
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1002 |
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1003 \begin{example} |
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1004 Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
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1005 \end{example} |
1001 |
1006 |
1002 \subsection{Modules as boundary labels} |
1007 \subsection{Modules as boundary labels} |
1003 \label{moddecss} |
1008 \label{moddecss} |
1004 |
1009 |
1005 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1010 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$), |
1006 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1011 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1007 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1012 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1008 |
1013 |
1009 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1014 %Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
1010 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
1015 %and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |