365 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
365 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
366 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
366 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
367 for any homeomorphic pair $X$ and $Y$, |
367 for any homeomorphic pair $X$ and $Y$, |
368 satisfying corresponding conditions. |
368 satisfying corresponding conditions. |
369 |
369 |
370 \nn{KW: the next paragraph seems awkward to me} |
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371 |
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372 \nn{KW: also, I'm not convinced that all of these (above and below) should be called theorems} |
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373 |
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374 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
370 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
375 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
371 Below, when we talk about the blob complex for a topological $n$-category, we are implicitly passing first to this associated system of fields. |
376 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
372 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. In that section we describe how to use the blob complex to construct $A_\infty$ $n$-categories from topological $n$-categories: |
377 |
373 |
378 \todo{Give this a number inside the text} |
374 \newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}} |
379 \begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
375 |
380 \label{thm:blobs-ainfty} |
376 \begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
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377 %\label{thm:blobs-ainfty} |
381 Let $\cC$ be a topological $n$-category. |
378 Let $\cC$ be a topological $n$-category. |
382 Let $Y$ be an $n{-}k$-manifold. |
379 Let $Y$ be an $n{-}k$-manifold. |
383 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
380 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
384 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
381 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
385 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
382 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
386 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
383 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
387 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
384 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
388 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
385 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
389 \end{thm} |
386 \end{ex:blob-complexes-of-balls} |
390 \begin{rem} |
387 \begin{rem} |
391 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. |
388 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. |
392 We think of this $A_\infty$ $n$-category as a free resolution. |
389 We think of this $A_\infty$ $n$-category as a free resolution. |
393 \end{rem} |
390 \end{rem} |
394 |
391 |
395 Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}. |
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396 |
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397 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
392 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
398 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
393 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
399 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
394 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
400 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
395 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
401 |
396 |
402 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
397 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
403 |
398 |
404 \begin{thm:product}[Product formula] |
399 \begin{thm:product}[Product formula] |
405 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
400 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
406 Let $\cC$ be an $n$-category. |
401 Let $\cC$ be an $n$-category. |
407 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Theorem \ref{thm:blobs-ainfty}). |
402 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). |
408 Then |
403 Then |
409 \[ |
404 \[ |
410 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
405 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
411 \] |
406 \] |
412 \end{thm:product} |
407 \end{thm:product} |