| changeset 225 | 32a76e8886d1 |
| parent 224 | 9faf1f7fad3e |
| child 236 | 3feb6e24a518 |
| 224:9faf1f7fad3e | 225:32a76e8886d1 |
|---|---|
555 \rm |
555 \rm |
556 \label{ex:bordism-category} |
556 \label{ex:bordism-category} |
557 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
557 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
558 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
558 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
559 to $\bd X$. |
559 to $\bd X$. |
560 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
560 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
561 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
561 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
562 $W \to W'$ which restricts to the identity on the boundary. |
562 $W \to W'$ which restricts to the identity on the boundary. |
563 \end{example} |
563 \end{example} |
564 |
564 |
565 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
565 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
612 \subsection{From $n$-categories to systems of fields} |
612 \subsection{From $n$-categories to systems of fields} |
613 \label{ss:ncat_fields} |
613 \label{ss:ncat_fields} |
614 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds. |
614 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variety) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds. |
615 In the case of plain $n$-categories, this is just the usual construction of a TQFT |
615 In the case of plain $n$-categories, this is just the usual construction of a TQFT |
616 from an $n$-category. |
616 from an $n$-category. |
617 For $\infty$ $n$-categories \nn{or whatever we decide to call them}, this gives an alternate (and |
617 For $A_\infty$ $n$-categories, this gives an alternate (and |
618 somewhat more canonical/tautological) construction of the blob complex. |
618 somewhat more canonical/tautological) construction of the blob complex. |
619 \nn{though from this point of view it seems more natural to just add some |
619 \nn{though from this point of view it seems more natural to just add some |
620 adjective to ``TQFT" rather than coining a completely new term like ``blob complex".} |
620 adjective to ``TQFT" rather than coining a completely new term like ``blob complex".} |
621 |
621 |
622 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
622 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
695 \end{equation*} |
695 \end{equation*} |
696 where $K$ is the vector space spanned by elements $a - g(a)$, with |
696 where $K$ is the vector space spanned by elements $a - g(a)$, with |
697 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
697 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
698 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
698 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
699 |
699 |
700 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit |
700 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
701 is more involved. |
701 is more involved. |
702 %\nn{should probably rewrite this to be compatible with some standard reference} |
702 %\nn{should probably rewrite this to be compatible with some standard reference} |
703 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
703 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
704 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
704 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$. |
705 Define $V$ as a vector space via |
705 Define $V$ as a vector space via |
993 |
993 |
994 \medskip |
994 \medskip |
995 |
995 |
996 We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
996 We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
997 |
997 |
998 Examples of modules: |
998 \begin{example}[Examples from TQFTs] |
999 \begin{itemize} |
999 \todo{} |
1000 \item \nn{examples from TQFTs} |
1000 \end{example} |
1001 \end{itemize} |
|
1002 |
1001 |
1003 \begin{example} |
1002 \begin{example} |
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. |
1003 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. |
1005 \end{example} |
1004 \end{example} |
1006 |
1005 |