29 |
29 |
30 \subsection{A product formula} |
30 \subsection{A product formula} |
31 \label{ss:product-formula} |
31 \label{ss:product-formula} |
32 |
32 |
33 |
33 |
34 Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from |
34 Given an $n$-dimensional system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from |
35 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ |
35 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ |
36 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
36 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
37 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$. |
37 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$. |
38 |
38 |
39 |
39 |
198 are homotopic. |
198 are homotopic. |
199 |
199 |
200 This concludes the proof of Theorem \ref{thm:product}. |
200 This concludes the proof of Theorem \ref{thm:product}. |
201 \end{proof} |
201 \end{proof} |
202 |
202 |
203 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
203 %\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
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204 |
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205 If $Y$ has dimension $k-m$, then we have an $m$-category $\cC_{Y\times F}$ whose value at |
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206 a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$ |
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207 (if $j=m$). |
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208 (See Example \ref{ex:blob-complexes-of-balls}.) |
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209 Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$. |
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210 These two categories are equivalent, but since we do not define functors between |
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211 topological $n$-categories in this paper we are unable to say precisely |
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212 what ``equivalent" means in this context. |
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213 We hope to include this stronger result in a future paper. |
204 |
214 |
205 \medskip |
215 \medskip |
206 |
216 |
207 Taking $F$ above to be a point, we obtain the following corollary. |
217 Taking $F$ in Theorem \ref{thm:product} to be a point, we obtain the following corollary. |
208 |
218 |
209 \begin{cor} |
219 \begin{cor} |
210 \label{cor:new-old} |
220 \label{cor:new-old} |
211 Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ |
221 Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ |
212 $n$-category obtained from $\cE$ by taking the blob complex of balls. |
222 $n$-category obtained from $\cE$ by taking the blob complex of balls. |
322 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
332 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
323 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
333 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
324 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
334 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
325 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
335 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
326 (See Example \ref{bc-module-example}.) |
336 (See Example \ref{bc-module-example}.) |
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337 \item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is |
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338 an $A_\infty$ $n{-}k$-category. |
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339 (See \S \ref{moddecss}.) |
327 \end{itemize} |
340 \end{itemize} |
328 |
341 |
329 \nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe |
342 It is the case that the $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$ |
330 just say that until we define functors we can't do more} |
343 are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories |
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344 in this paper, we cannot state this precisely. |
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345 (It will appear in a future paper.) |
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346 So we content ourselves with |
331 |
347 |
332 \begin{thm} |
348 \begin{thm} |
333 \label{thm:gluing} |
349 \label{thm:gluing} |
334 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
350 When $k=n$ above, $\bc(X)$ is homotopy equivalent to $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
335 \end{thm} |
351 \end{thm} |
336 |
352 |
337 \begin{proof} |
353 \begin{proof} |
338 We will assume $k=n$; the other cases are similar. |
354 %We will assume $k=n$; the other cases are similar. |
339 The proof is similar to that of Theorem \ref{thm:product}. |
355 The proof is similar to that of Theorem \ref{thm:product}. |
340 We give a short sketch with emphasis on the differences from |
356 We give a short sketch with emphasis on the differences from |
341 the proof of Theorem \ref{thm:product}. |
357 the proof of Theorem \ref{thm:product}. |
342 |
358 |
343 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
359 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |