532 to be the dual Hilbert space $A(X\times F; c)$. |
532 to be the dual Hilbert space $A(X\times F; c)$. |
533 \nn{refer elsewhere for details?} |
533 \nn{refer elsewhere for details?} |
534 \end{example} |
534 \end{example} |
535 |
535 |
536 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
536 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
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537 |
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538 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
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539 |
537 \newcommand{\Bord}{\operatorname{Bord}} |
540 \newcommand{\Bord}{\operatorname{Bord}} |
538 \begin{example}[The bordism $n$-category] |
541 \begin{example}[The bordism $n$-category] |
539 \rm |
542 \rm |
540 \label{ex:bordism-category} |
543 \label{ex:bordism-category} |
541 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
544 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
593 |
596 |
594 |
597 |
595 \subsection{From $n$-categories to systems of fields} |
598 \subsection{From $n$-categories to systems of fields} |
596 \label{ss:ncat_fields} |
599 \label{ss:ncat_fields} |
597 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. |
600 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. |
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601 In the case of plain $n$-categories, this is just the usual construction of a TQFT |
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602 from an $n$-category. |
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603 For $\infty$ $n$-categories \nn{or whatever we decide to call them}, this gives an alternate (and |
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604 somewhat more canonical/tautological) construction of the blob complex. |
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605 \nn{though from this point of view it seems more natural to just add some |
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606 adjective to ``TQFT" rather than coining a completely new term like ``blob complex".} |
598 |
607 |
599 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
608 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
600 An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
609 An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we will define $\cC(W)$ as a suitable colimit (or homotopy colimit in the $A_\infty$ case) of this functor. |
601 We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting system of fields is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
610 We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data), then the resulting system of fields is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
602 |
611 |
1092 |
1101 |
1093 |
1102 |
1094 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1103 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
1095 a separate paper): |
1104 a separate paper): |
1096 \begin{itemize} |
1105 \begin{itemize} |
1097 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
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1098 \item conversely, our def implies other defs |
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1099 \item do same for modules; maybe an appendix on relating topological |
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1100 vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
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1101 \item traditional $A_\infty$ 1-cat def implies our def |
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1102 \item ... and vice-versa (already done in appendix) |
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1103 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
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1104 \item spell out what difference (if any) Top vs PL vs Smooth makes |
1106 \item spell out what difference (if any) Top vs PL vs Smooth makes |
1105 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
1107 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
1106 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
1108 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
1107 \item morphisms of modules; show that it's adjoint to tensor product |
1109 \item morphisms of modules; show that it's adjoint to tensor product |
1108 (need to define dual module for this) |
1110 (need to define dual module for this) |
1109 \item functors |
1111 \item functors |
1110 \end{itemize} |
1112 \end{itemize} |
1111 |
1113 |
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1114 \bigskip |
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1115 |
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1116 \hrule |
1112 \nn{Some salvaged paragraphs that we might want to work back in:} |
1117 \nn{Some salvaged paragraphs that we might want to work back in:} |
1113 \hrule |
1118 \bigskip |
1114 |
1119 |
1115 Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) |
1120 Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) |
1116 |
1121 |
1117 The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition |
1122 The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition |
1118 \begin{align*} |
1123 \begin{align*} |