356 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
356 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
357 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
357 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
358 for any homeomorphic pair $X$ and $Y$, |
358 for any homeomorphic pair $X$ and $Y$, |
359 satisfying corresponding conditions. |
359 satisfying corresponding conditions. |
360 |
360 |
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361 \nn{KW: the next paragraph seems awkward to me} |
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362 |
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363 \nn{KW: also, I'm not convinced that all of these (above and below) should be called theorems} |
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364 |
361 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
365 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
362 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
366 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
363 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
367 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
364 |
368 |
365 \todo{Give this a number inside the text} |
369 \todo{Give this a number inside the text} |
376 \end{thm} |
380 \end{thm} |
377 \begin{rem} |
381 \begin{rem} |
378 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. |
382 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. |
379 We think of this $A_\infty$ $n$-category as a free resolution. |
383 We think of this $A_\infty$ $n$-category as a free resolution. |
380 \end{rem} |
384 \end{rem} |
381 Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats} |
385 |
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386 Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}. |
382 |
387 |
383 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
388 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
384 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
389 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
385 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
390 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
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391 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
386 |
392 |
387 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
393 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
388 |
394 |
389 \begin{thm:product}[Product formula] |
395 \begin{thm:product}[Product formula] |
390 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
396 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
393 Then |
399 Then |
394 \[ |
400 \[ |
395 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
401 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
396 \] |
402 \] |
397 \end{thm:product} |
403 \end{thm:product} |
398 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
404 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
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405 (see \S \ref{moddecss}). |
399 |
406 |
400 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
407 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
401 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
408 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
402 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
409 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
403 |
410 |
415 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
422 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
416 \end{equation*} |
423 \end{equation*} |
417 \end{itemize} |
424 \end{itemize} |
418 \end{thm:gluing} |
425 \end{thm:gluing} |
419 |
426 |
420 Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. |
427 Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}. |
421 |
428 |
422 \subsection{Applications} |
429 \subsection{Applications} |
423 \label{sec:applications} |
430 \label{sec:applications} |
424 Finally, we give two theorems which we consider as applications. |
431 Finally, we give two theorems which we consider applications. % or "think of as" |
425 |
432 |
426 \newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
433 \newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
427 |
434 |
428 \begin{thm:map-recon}[Mapping spaces] |
435 \begin{thm:map-recon}[Mapping spaces] |
429 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
436 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
430 $B^n \to T$. |
437 $B^n \to T$. |
431 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
438 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
432 Then |
439 Then |
433 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
440 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
434 \end{thm:map-recon} |
441 \end{thm:map-recon} |
435 |
442 |
436 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}. |
443 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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444 The proof appears in \S \ref{sec:map-recon}. |
437 |
445 |
438 \newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
446 \newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
439 |
447 |
440 \begin{thm:deligne}[Higher dimensional Deligne conjecture] |
448 \begin{thm:deligne}[Higher dimensional Deligne conjecture] |
441 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
449 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
442 \end{thm:deligne} |
450 \end{thm:deligne} |
443 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
451 See \S \ref{sec:deligne} for a full explanation of the statement, and the proof. |
444 |
452 |
445 |
453 |
446 |
454 |
447 |
455 |
448 \subsection{Future directions} |
456 \subsection{Future directions} |
449 \label{sec:future} |
457 \label{sec:future} |
450 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
458 \nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.} |
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459 Throughout, we have resisted the temptation to work in the greatest generality possible. |
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460 (Don't worry, it wasn't that hard.) |
451 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
461 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
452 We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), |
462 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
453 and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
463 %%%%%% |
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464 And likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
454 More could be said about finite characteristic |
465 More could be said about finite characteristic |
455 (there appears in be $2$-torsion in $\bc_1(S^2; \cC)$ for any spherical $2$-category $\cC$, for example). |
466 (there appears in be $2$-torsion in $\bc_1(S^2; \cC)$ for any spherical $2$-category $\cC$, for example). |
456 Much more could be said about other types of manifolds, in particular oriented, |
467 Much more could be said about other types of manifolds, in particular oriented, |
457 $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
468 $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
458 (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
469 (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |