368 for any homeomorphic pair $X$ and $Y$, |
372 for any homeomorphic pair $X$ and $Y$, |
369 satisfying corresponding conditions. |
373 satisfying corresponding conditions. |
370 |
374 |
371 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, |
375 In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, |
372 from which we can construct systems of fields. |
376 from which we can construct systems of fields. |
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377 Traditional $n$-categories can be converted to disk-like $n$-categories by taking string diagrams |
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378 (see \S\ref{sec:example:traditional-n-categories(fields)}). |
373 Below, when we talk about the blob complex for a disk-like $n$-category, |
379 Below, when we talk about the blob complex for a disk-like $n$-category, |
374 we are implicitly passing first to this associated system of fields. |
380 we are implicitly passing first to this associated system of fields. |
375 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
381 Further, in \S \ref{sec:ncats} we also have the notion of a disk-like $A_\infty$ $n$-category. |
376 In that section we describe how to use the blob complex to |
382 In that section we describe how to use the blob complex to |
377 construct $A_\infty$ $n$-categories from ordinary $n$-categories: |
383 construct disk-like $A_\infty$ $n$-categories from ordinary disk-like $n$-categories: |
378 |
384 |
379 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
385 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
380 |
386 |
381 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
387 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form a disk-like $A_\infty$ $n$-category] |
382 %\label{thm:blobs-ainfty} |
388 %\label{thm:blobs-ainfty} |
383 Let $\cC$ be an ordinary $n$-category. |
389 Let $\cC$ be an ordinary disk-like $n$-category. |
384 Let $Y$ be an $n{-}k$-manifold. |
390 Let $Y$ be an $n{-}k$-manifold. |
385 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
391 There is a disk-like $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
386 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
392 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
387 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
393 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
388 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
394 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
389 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
395 These sets have the structure of a disk-like $A_\infty$ $k$-category, with compositions coming from the gluing map in |
390 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
396 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
391 \end{ex:blob-complexes-of-balls} |
397 \end{ex:blob-complexes-of-balls} |
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398 |
392 \begin{rem} |
399 \begin{rem} |
393 Perhaps the most interesting case is when $Y$ is just a point; |
400 Perhaps the most interesting case is when $Y$ is just a point; |
394 then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category. |
401 then we have a way of building a disk-like $A_\infty$ $n$-category from an ordinary $n$-category. % disk-like or not |
395 We think of this $A_\infty$ $n$-category as a free resolution. |
402 We think of this disk-like $A_\infty$ $n$-category as a free resolution of the ordinary $n$-category. |
396 \end{rem} |
403 \end{rem} |
397 |
404 |
398 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
405 There is a version of the blob complex for $\cC$ a disk-like $A_\infty$ $n$-category |
399 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
406 instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}. |
400 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
407 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
401 The next theorem describes the blob complex for product manifolds, |
408 The next theorem describes the blob complex for product manifolds |
402 in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
409 in terms of the $A_\infty$ blob complex of the disk-like $A_\infty$ $n$-categories constructed as in the previous example. |
403 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
410 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
404 |
411 |
405 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
412 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
406 |
413 |
407 \begin{thm:product}[Product formula] |
414 \begin{thm:product}[Product formula] |
408 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
415 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
409 Let $\cC$ be an $n$-category. |
416 Let $\cC$ be an $n$-category. |
410 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology |
417 Let $\bc_*(Y;\cC)$ be the disk-like $A_\infty$ $k$-category associated to $Y$ via blob homology |
411 (see Example \ref{ex:blob-complexes-of-balls}). |
418 (see Example \ref{ex:blob-complexes-of-balls}). |
412 Then |
419 Then |
413 \[ |
420 \[ |
414 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
421 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
415 \] |
422 \] |
416 \end{thm:product} |
423 \end{thm:product} |
417 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
424 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
418 (see \S \ref{ss:product-formula}). |
425 (see \S \ref{ss:product-formula}). |
419 |
426 |
420 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
427 Fix a disk-like $n$-category $\cC$, which we'll omit from the notation. |
421 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
428 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ 1-category. |
422 (See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
429 (See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories |
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430 and the usual algebraic notion of an $A_\infty$ category.) |
423 |
431 |
424 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
432 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
425 |
433 |
426 \begin{thm:gluing}[Gluing formula] |
434 \begin{thm:gluing}[Gluing formula] |
427 \mbox{}% <-- gets the indenting right |
435 \mbox{}% <-- gets the indenting right |
428 \begin{itemize} |
436 \begin{itemize} |
429 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
437 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
430 $A_\infty$ module for $\bc_*(Y)$. |
438 $A_\infty$ module for $\bc_*(Y)$. |
431 |
439 |
432 \item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of |
440 \item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ |
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441 is the $A_\infty$ self-tensor product of |
433 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
442 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
434 \begin{equation*} |
443 \begin{equation*} |
435 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
444 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
436 \end{equation*} |
445 \end{equation*} |
437 \end{itemize} |
446 \end{itemize} |