619 \begin{example}[Traditional $n$-categories] |
619 \begin{example}[Traditional $n$-categories] |
620 \rm |
620 \rm |
621 \label{ex:traditional-n-categories} |
621 \label{ex:traditional-n-categories} |
622 Given a `traditional $n$-category with strong duality' $C$ |
622 Given a `traditional $n$-category with strong duality' $C$ |
623 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
623 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
624 to be the set of all $C$-labeled sub cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
624 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
625 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
625 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
626 combinations of $C$-labeled sub cell complexes of $X$ |
626 combinations of $C$-labeled embedded cell complexes of $X$ |
627 modulo the kernel of the evaluation map. |
627 modulo the kernel of the evaluation map. |
628 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
628 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
629 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$. |
629 with each cell labelled according to the corresponding cell for $a$. |
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630 (These two cells have the same codimension.) |
630 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
631 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
631 Define $\cC(X)$, for $\dim(X) < n$, |
632 Define $\cC(X)$, for $\dim(X) < n$, |
632 to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
633 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
633 Define $\cC(X; c)$, for $X$ an $n$-ball, |
634 Define $\cC(X; c)$, for $X$ an $n$-ball, |
634 to be the dual Hilbert space $A(X\times F; c)$. |
635 to be the dual Hilbert space $A(X\times F; c)$. |
635 \nn{refer elsewhere for details?} |
636 \nn{refer elsewhere for details?} |
636 |
637 |
637 |
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638 Recall we described a system of fields and local relations based on a `traditional $n$-category' |
638 Recall we described a system of fields and local relations based on a `traditional $n$-category' |
639 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
639 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
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640 \nn{KW: We already refer to \S \ref{sec:fields} above} |
640 Constructing a system of fields from $\cC$ recovers that example. |
641 Constructing a system of fields from $\cC$ recovers that example. |
641 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
642 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
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643 \nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
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644 where the quotient is built in. |
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645 but (string diagrams)/(relations) is isomorphic to |
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646 (pasting diagrams composed of smaller string diagrams)/(relations)} |
642 \end{example} |
647 \end{example} |
643 |
648 |
644 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
649 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
645 |
650 |
646 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
651 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
696 \end{example} |
701 \end{example} |
697 |
702 |
698 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. |
703 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. |
699 Notice that with $F$ a point, the above example is a construction turning a topological |
704 Notice that with $F$ a point, the above example is a construction turning a topological |
700 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
705 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
701 We think of this as providing a `free resolution' of the topological $n$-category. |
706 We think of this as providing a `free resolution' |
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707 \nn{`cofibrant replacement'?} |
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708 of the topological $n$-category. |
702 \todo{Say more here!} |
709 \todo{Say more here!} |
703 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
710 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
704 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
711 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
705 and take $\CD{B}$ to act trivially. |
712 and take $\CD{B}$ to act trivially. |
706 |
713 |
714 \label{ex:bordism-category-ainf} |
721 \label{ex:bordism-category-ainf} |
715 blah blah \nn{to do...} |
722 blah blah \nn{to do...} |
716 \end{example} |
723 \end{example} |
717 |
724 |
718 |
725 |
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726 |
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727 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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728 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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729 (We require that the interiors of the little balls be disjoint, but their |
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730 boundaries are allowed to meet. |
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731 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
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732 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
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733 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
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734 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
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735 (By shrinking the little balls (precomposing them with dilations), |
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736 we see that both operads are homotopic to the space of $k$ framed points |
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737 in $B^n$.) |
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738 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have the structure have |
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739 an action of $\cE\cB_n$. |
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740 \nn{add citation for this operad if we can find one} |
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741 |
719 \begin{example}[$E_n$ algebras] |
742 \begin{example}[$E_n$ algebras] |
720 \rm |
743 \rm |
721 \label{ex:e-n-alg} |
744 \label{ex:e-n-alg} |
722 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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723 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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724 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
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725 (By peeling the little balls, we see that both are homotopic to the space of $k$ framed points |
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726 in $B^n$.) |
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727 |
745 |
728 Let $A$ be an $\cE\cB_n$-algebra. |
746 Let $A$ be an $\cE\cB_n$-algebra. |
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747 Note that this implies a $\Diff(B^n)$ action on $A$, |
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748 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
729 We will define an $A_\infty$ $n$-category $\cC^A$. |
749 We will define an $A_\infty$ $n$-category $\cC^A$. |
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750 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
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751 In other words, the $k$-morphisms are trivial for $k<n$. |
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752 %If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
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753 %(Plain colimit, not homotopy colimit.) |
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754 %Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
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755 %the standard ball $B^n$ into $X$, and who morphisms are given by engu |
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756 |
730 \nn{...} |
757 \nn{...} |
731 \end{example} |
758 \end{example} |
732 |
759 |
733 |
760 |
734 |
761 |
1141 \medskip |
1168 \medskip |
1142 |
1169 |
1143 Note that the above axioms imply that an $n$-category module has the structure |
1170 Note that the above axioms imply that an $n$-category module has the structure |
1144 of an $n{-}1$-category. |
1171 of an $n{-}1$-category. |
1145 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1172 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
1146 where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
1173 where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
1147 above the non-marked boundary component of $J$. |
1174 above the non-marked boundary component of $J$. |
1148 (More specifically, we collapse $X\times P$ to a single point, where |
1175 (More specifically, we collapse $X\times P$ to a single point, where |
1149 $P$ is the non-marked boundary component of $J$.) |
1176 $P$ is the non-marked boundary component of $J$.) |
1150 \nn{give figure for this?} |
1177 \nn{give figure for this?} |
1151 Then $\cE$ has the structure of an $n{-}1$-category. |
1178 Then $\cE$ has the structure of an $n{-}1$-category. |