33 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (Theorem \ref{thm:evaluation}) and a gluing |
33 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (Theorem \ref{thm:evaluation}) and a gluing |
34 formula allowing calculations by cutting manifolds into smaller parts (Theorem \ref{thm:gluing}). |
34 formula allowing calculations by cutting manifolds into smaller parts (Theorem \ref{thm:gluing}). |
35 |
35 |
36 We expect applications of the blob complex to contact topology and Khovanov homology |
36 We expect applications of the blob complex to contact topology and Khovanov homology |
37 but do not address these in this paper. |
37 but do not address these in this paper. |
38 See \S \ref{sec:future} for slightly more detail. |
38 %See \S \ref{sec:future} for slightly more detail. |
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39 |
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40 Throughout, we have resisted the temptation to work in the greatest possible generality. |
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41 (Don't worry, it wasn't that hard.) |
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42 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category |
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43 with sufficient limits and colimits would do. |
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44 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
39 |
45 |
40 |
46 |
41 \subsection{Structure of the paper} |
47 \subsection{Structure of the paper} |
42 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
48 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
43 summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) |
49 summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}). |
44 and outline anticipated future directions (see \S \ref{sec:future}). |
50 %and outline anticipated future directions (see \S \ref{sec:future}). |
45 \nn{recheck this list after done editing intro} |
51 %\nn{recheck this list after done editing intro} |
46 |
52 |
47 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
53 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
48 and establishes some of its properties. |
54 and establishes some of its properties. |
49 There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is |
55 There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is |
50 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
56 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
453 \end{thm:deligne} |
459 \end{thm:deligne} |
454 See \S \ref{sec:deligne} for a full explanation of the statement, and the proof. |
460 See \S \ref{sec:deligne} for a full explanation of the statement, and the proof. |
455 |
461 |
456 |
462 |
457 |
463 |
458 |
464 \noop{ |
459 \subsection{Future directions} |
465 \subsection{Future directions} |
460 \label{sec:future} |
466 \label{sec:future} |
461 \nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.} |
467 \nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.} |
462 Throughout, we have resisted the temptation to work in the greatest generality possible. |
468 Throughout, we have resisted the temptation to work in the greatest generality possible. |
463 (Don't worry, it wasn't that hard.) |
469 (Don't worry, it wasn't that hard.) |
480 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
486 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
481 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
487 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
482 but haven't investigated the details. |
488 but haven't investigated the details. |
483 |
489 |
484 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
490 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
485 |
491 } %%% end \noop |
486 |
492 |
487 \subsection{Thanks and acknowledgements} |
493 \subsection{Thanks and acknowledgements} |
488 % attempting to make this chronological rather than alphabetical |
494 % attempting to make this chronological rather than alphabetical |
489 We'd like to thank |
495 We'd like to thank |
490 Justin Roberts, |
496 Justin Roberts, |
497 Kevin Costello, |
503 Kevin Costello, |
498 Chris Douglas, |
504 Chris Douglas, |
499 and |
505 and |
500 Alexander Kirillov |
506 Alexander Kirillov |
501 for many interesting and useful conversations. |
507 for many interesting and useful conversations. |
502 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the conducive environment provided there during the final preparation of this manuscript. |
508 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive |
503 |
509 % "conducive" needs an object; "conducive to blah" |
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510 environment provided there during the final preparation of this manuscript. |
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511 |