236 F \to E \to Y . |
236 F \to E \to Y . |
237 \] |
237 \] |
238 We outline one approach here and a second in Subsection xxxx. |
238 We outline one approach here and a second in Subsection xxxx. |
239 |
239 |
240 We can generalize the definition of a $k$-category by replacing the categories |
240 We can generalize the definition of a $k$-category by replacing the categories |
241 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$. |
241 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
242 \nn{need citation to other work that does this; Stolz and Teichner?} |
242 (c.f. \cite{MR2079378}). |
243 Call this a $k$-category over $Y$. |
243 Call this a $k$-category over $Y$. |
244 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
244 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
245 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
245 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
246 Let $\cF_E$ denote this $k$-category over $Y$. |
246 Let $\cF_E$ denote this $k$-category over $Y$. |
247 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
247 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
274 |
274 |
275 Next we prove a gluing theorem. |
275 Next we prove a gluing theorem. |
276 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
276 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
277 We will need an explicit collar on $Y$, so rewrite this as |
277 We will need an explicit collar on $Y$, so rewrite this as |
278 $X = X_1\cup (Y\times J) \cup X_2$. |
278 $X = X_1\cup (Y\times J) \cup X_2$. |
279 Given this data we have: \nn{need refs to above for these} |
279 Given this data we have: |
280 \begin{itemize} |
280 \begin{itemize} |
281 \item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
281 \item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
282 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
282 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
283 (for $m+k = n$). \nn{need to explain $c$}. |
283 (for $m+k = n$). |
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284 (See Example \ref{ex:blob-complexes-of-balls}.) |
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285 %\nn{need to explain $c$}. |
284 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
286 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
285 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
287 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
286 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
288 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
287 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
289 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
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290 (See Example \nn{need example for this}.) |
288 \end{itemize} |
291 \end{itemize} |
289 |
292 |
290 \begin{thm} |
293 \begin{thm} |
291 \label{thm:gluing} |
294 \label{thm:gluing} |
292 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
295 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
293 \end{thm} |
296 \end{thm} |
294 |
297 |
295 \begin{proof} |
298 \begin{proof} |
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299 \nn{for now, just prove $k=0$ case.} |
296 The proof is similar to that of Theorem \ref{product_thm}. |
300 The proof is similar to that of Theorem \ref{product_thm}. |
297 \nn{need to say something about dimensions less than $n$, |
301 We give a short sketch with emphasis on the differences from |
298 but for now concentrate on top dimension.} |
302 the proof of Theorem \ref{product_thm}. |
299 |
303 |
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304 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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305 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
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306 |
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307 We define a map $\psi:\cT\to \bc_*(X)$. On filtration degree zero summands it is given |
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308 by gluing the pieces together to get a blob diagram on $X$. |
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309 On filtration degree 1 and greater $\psi$ is zero. |
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310 |
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311 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
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312 over some decomposition of $J$. |
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313 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
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314 a subcomplex of $G_*$. |
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315 |
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316 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
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317 As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$ |
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318 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
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319 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
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320 a common refinement. |
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321 |
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322 The proof that these two maps are inverse to each other is the same as in |
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323 Theorem \ref{product_thm}. |
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324 \end{proof} |
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325 |
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326 This establishes Property \ref{property:gluing}. |
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327 |
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328 \noop{ |
300 Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
329 Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
301 Let $D$ be an $n{-}k$-ball. |
330 Let $D$ be an $n{-}k$-ball. |
302 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
331 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
303 To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex |
332 To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex |
304 $\cS_*$ which is adapted to a fine open cover of $D\times X$. |
333 $\cS_*$ which is adapted to a fine open cover of $D\times X$. |
305 For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ |
334 For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ |
306 on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding |
335 on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding |
307 decomposition of $D\times X$. |
336 decomposition of $D\times X$. |
308 The proof that these two maps are inverse to each other is the same as in |
337 The proof that these two maps are inverse to each other is the same as in |
309 Theorem \ref{product_thm}. |
338 Theorem \ref{product_thm}. |
310 \end{proof} |
339 } |
311 |
340 |
312 This establishes Property \ref{property:gluing}. |
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313 |
341 |
314 \medskip |
342 \medskip |
315 |
343 |
316 \subsection{Reconstructing mapping spaces} |
344 \subsection{Reconstructing mapping spaces} |
317 |
345 |