40 for $X$ an $m$-ball with $m\le k$. |
40 for $X$ an $m$-ball with $m\le k$. |
41 } |
41 } |
42 |
42 |
43 \nn{need to settle on notation; proof and statement are inconsistent} |
43 \nn{need to settle on notation; proof and statement are inconsistent} |
44 |
44 |
45 \begin{thm} \label{product_thm} |
45 \begin{thm} \label{thm:product} |
46 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
46 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
47 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
47 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
48 \begin{equation*} |
48 \begin{equation*} |
49 C^{\times F}(B) = \cB_*(B \times F, C). |
49 C^{\times F}(B) = \cB_*(B \times F, C). |
50 \end{equation*} |
50 \end{equation*} |
55 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
55 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
56 \end{align*} |
56 \end{align*} |
57 \end{thm} |
57 \end{thm} |
58 |
58 |
59 |
59 |
60 \begin{proof}%[Proof of Theorem \ref{product_thm}] |
60 \begin{proof} |
61 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
61 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
62 |
62 |
63 First we define a map |
63 First we define a map |
64 \[ |
64 \[ |
65 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
65 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
212 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
212 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
213 Both the identity map and $\phi\circ\psi$ are compatible with this |
213 Both the identity map and $\phi\circ\psi$ are compatible with this |
214 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps |
214 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps |
215 are homotopic. |
215 are homotopic. |
216 |
216 |
217 This concludes the proof of Theorem \ref{product_thm}. |
217 This concludes the proof of Theorem \ref{thm:product}. |
218 \end{proof} |
218 \end{proof} |
219 |
219 |
220 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
220 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
221 |
221 |
222 \medskip |
222 \medskip |
245 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
245 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
246 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
246 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
247 Let $\cF_E$ denote this $k$-category over $Y$. |
247 Let $\cF_E$ denote this $k$-category over $Y$. |
248 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
248 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
249 get a chain complex $\cF_E(Y)$. |
249 get a chain complex $\cF_E(Y)$. |
250 The proof of Theorem \ref{product_thm} goes through essentially unchanged |
250 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
251 to show that |
251 to show that |
252 \[ |
252 \[ |
253 \bc_*(E) \simeq \cF_E(Y) . |
253 \bc_*(E) \simeq \cF_E(Y) . |
254 \] |
254 \] |
255 |
255 |
296 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
296 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
297 \end{thm} |
297 \end{thm} |
298 |
298 |
299 \begin{proof} |
299 \begin{proof} |
300 \nn{for now, just prove $k=0$ case.} |
300 \nn{for now, just prove $k=0$ case.} |
301 The proof is similar to that of Theorem \ref{product_thm}. |
301 The proof is similar to that of Theorem \ref{thm:product}. |
302 We give a short sketch with emphasis on the differences from |
302 We give a short sketch with emphasis on the differences from |
303 the proof of Theorem \ref{product_thm}. |
303 the proof of Theorem \ref{thm:product}. |
304 |
304 |
305 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
305 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
306 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
306 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
307 |
307 |
308 We define a map $\psi:\cT\to \bc_*(X)$. |
308 We define a map $\psi:\cT\to \bc_*(X)$. |
314 over some decomposition of $J$. |
314 over some decomposition of $J$. |
315 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
315 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
316 a subcomplex of $G_*$. |
316 a subcomplex of $G_*$. |
317 |
317 |
318 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
318 Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
319 As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$ |
319 As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
320 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
320 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
321 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
321 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
322 a common refinement. |
322 a common refinement. |
323 |
323 |
324 The proof that these two maps are inverse to each other is the same as in |
324 The proof that these two maps are inverse to each other is the same as in |
325 Theorem \ref{product_thm}. |
325 Theorem \ref{thm:product}. |
326 \end{proof} |
326 \end{proof} |
327 |
|
328 This establishes Property \ref{property:gluing}. |
|
329 |
327 |
330 \noop{ |
328 \noop{ |
331 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 $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
332 Let $D$ be an $n{-}k$-ball. |
330 Let $D$ be an $n{-}k$-ball. |
333 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
331 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
335 $\cS_*$ which is adapted to a fine open cover of $D\times X$. |
333 $\cS_*$ which is adapted to a fine open cover of $D\times X$. |
336 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 $b$ |
337 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 |
338 decomposition of $D\times X$. |
336 decomposition of $D\times X$. |
339 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 |
340 Theorem \ref{product_thm}. |
338 Theorem \ref{thm:product}. |
341 } |
339 } |
342 |
340 |
343 |
341 |
344 \medskip |
342 \medskip |
345 |
343 |
346 \subsection{Reconstructing mapping spaces} |
344 \subsection{Reconstructing mapping spaces} |
|
345 \label{sec:map-recon} |
347 |
346 |
348 The next theorem shows how to reconstruct a mapping space from local data. |
347 The next theorem shows how to reconstruct a mapping space from local data. |
349 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
348 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
350 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
349 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
351 of Example \ref{ex:chains-of-maps-to-a-space}. |
350 of Example \ref{ex:chains-of-maps-to-a-space}. |
352 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
351 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
353 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
352 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
354 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
353 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
355 |
354 |
356 \begin{thm} \label{thm:map-recon} |
355 \begin{thm} |
|
356 \label{thm:map-recon} |
357 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
357 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
358 is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
358 is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
359 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
359 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
360 \end{thm} |
360 \end{thm} |
361 \begin{rem} |
361 \begin{rem} |
367 is trivial at all but the topmost level. |
367 is trivial at all but the topmost level. |
368 Ricardo Andrade also told us about a similar result. |
368 Ricardo Andrade also told us about a similar result. |
369 \end{rem} |
369 \end{rem} |
370 |
370 |
371 \begin{proof} |
371 \begin{proof} |
372 The proof is again similar to that of Theorem \ref{product_thm}. |
372 The proof is again similar to that of Theorem \ref{thm:product}. |
373 |
373 |
374 We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
374 We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
375 |
375 |
376 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
376 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
377 $j$-fold mapping cylinders, $j \ge 0$. |
377 $j$-fold mapping cylinders, $j \ge 0$. |
409 We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models. |
409 We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models. |
410 Let $a$ be a generator of $G_*$. |
410 Let $a$ be a generator of $G_*$. |
411 Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all |
411 Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all |
412 pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$ |
412 pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$ |
413 and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$. |
413 and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$. |
414 (See the proof of Theorem \ref{product_thm} for more details.) |
414 (See the proof of Theorem \ref{thm:product} for more details.) |
415 The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic. |
415 The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic. |
416 By the usual acyclic models nonsense, there is a (unique up to homotopy) |
416 By the usual acyclic models nonsense, there is a (unique up to homotopy) |
417 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
417 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
418 Furthermore, we may choose $\phi$ such that for all $a$ |
418 Furthermore, we may choose $\phi$ such that for all $a$ |
419 \[ |
419 \[ |
421 \] |
421 \] |
422 where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$. |
422 where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$. |
423 |
423 |
424 It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
424 It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
425 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
425 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
426 (See the proof of Theorem \ref{product_thm} for more details.) |
426 (See the proof of Theorem \ref{thm:product} for more details.) |
427 \end{proof} |
427 \end{proof} |
428 |
428 |
429 \noop{ |
429 \noop{ |
430 % old proof (just start): |
430 % old proof (just start): |
431 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
431 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |