206 |
206 |
207 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
207 \nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
208 |
208 |
209 \medskip |
209 \medskip |
210 |
210 |
|
211 Taking $F$ above to be a point, we obtain the following corollary. |
|
212 |
211 \begin{cor} |
213 \begin{cor} |
212 \label{cor:new-old} |
214 \label{cor:new-old} |
213 The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point: |
215 Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ |
214 $$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$ |
216 $n$-category obtained from $\cE$ by taking the blob complex of balls. |
|
217 Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are |
|
218 homotopy equivalent: |
|
219 \[ |
|
220 \bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) . |
|
221 \] |
215 \end{cor} |
222 \end{cor} |
216 \begin{proof} |
|
217 Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point. |
|
218 \end{proof} |
|
219 |
223 |
220 \medskip |
224 \medskip |
221 |
225 |
222 Theorem \ref{thm:product} extends to the case of general fiber bundles |
226 Theorem \ref{thm:product} extends to the case of general fiber bundles |
223 \[ |
227 \[ |
224 F \to E \to Y . |
228 F \to E \to Y . |
225 \] |
229 \] |
226 We outline one approach here and a second in Subsection xxxx. |
230 We outline one approach here and a second in \S \ref{xyxyx}. |
227 |
231 |
228 We can generalize the definition of a $k$-category by replacing the categories |
232 We can generalize the definition of a $k$-category by replacing the categories |
229 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
233 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
230 (c.f. \cite{MR2079378}). |
234 (c.f. \cite{MR2079378}). |
231 Call this a $k$-category over $Y$. |
235 Call this a $k$-category over $Y$. |
232 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
236 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
233 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
237 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
234 Let $\cF_E$ denote this $k$-category over $Y$. |
238 Let $\cF_E$ denote this $k$-category over $Y$. |
235 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
239 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
236 get a chain complex $\cF_E(Y)$. |
240 get a chain complex $\cl{\cF_E}(Y)$. |
237 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
241 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
238 to show that |
242 to show that |
239 \[ |
243 \[ |
240 \bc_*(E) \simeq \cF_E(Y) . |
244 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
241 \] |
245 \] |
242 |
246 |
243 \nn{remark further that this still works when the map is not even a fibration?} |
247 \nn{remark further that this still works when the map is not even a fibration?} |
244 |
248 |
245 \nn{put this later} |
249 \nn{put this later} |
274 %\nn{need to explain $c$}. |
278 %\nn{need to explain $c$}. |
275 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
279 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
276 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
280 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
277 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
281 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
278 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
282 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
279 (See Example \nn{need example for this}.) |
283 (See Example \ref{bc-module-example}.) |
280 \end{itemize} |
284 \end{itemize} |
|
285 |
|
286 \nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe |
|
287 just say that until we define functors we can't do more} |
281 |
288 |
282 \begin{thm} |
289 \begin{thm} |
283 \label{thm:gluing} |
290 \label{thm:gluing} |
284 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
291 $\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
285 \end{thm} |
292 \end{thm} |
286 |
293 |
287 \begin{proof} |
294 \begin{proof} |
288 \nn{for now, just prove $k=0$ case.} |
295 We will assume $k=n$; the other cases are similar. |
289 The proof is similar to that of Theorem \ref{thm:product}. |
296 The proof is similar to that of Theorem \ref{thm:product}. |
290 We give a short sketch with emphasis on the differences from |
297 We give a short sketch with emphasis on the differences from |
291 the proof of Theorem \ref{thm:product}. |
298 the proof of Theorem \ref{thm:product}. |
292 |
299 |
293 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
300 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
294 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
301 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
295 |
302 |
296 We define a map $\psi:\cT\to \bc_*(X)$. |
303 We define a map $\psi:\cT\to \bc_*(X)$. |
297 On filtration degree zero summands it is given |
304 On 0-simplices it is given |
298 by gluing the pieces together to get a blob diagram on $X$. |
305 by gluing the pieces together to get a blob diagram on $X$. |
299 On filtration degree 1 and greater $\psi$ is zero. |
306 On simplices of dimension 1 and greater $\psi$ is zero. |
300 |
307 |
301 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
308 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
302 over some decomposition of $J$. |
309 over some decomposition of $J$. |
303 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
310 It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
304 a subcomplex of $G_*$. |
311 a subcomplex of $G_*$. |
310 a common refinement. |
317 a common refinement. |
311 |
318 |
312 The proof that these two maps are inverse to each other is the same as in |
319 The proof that these two maps are inverse to each other is the same as in |
313 Theorem \ref{thm:product}. |
320 Theorem \ref{thm:product}. |
314 \end{proof} |
321 \end{proof} |
315 |
|
316 \noop{ |
|
317 Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
|
318 Let $D$ be an $n{-}k$-ball. |
|
319 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
|
320 To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex |
|
321 $\cS_*$ which is adapted to a fine open cover of $D\times X$. |
|
322 For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ |
|
323 on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding |
|
324 decomposition of $D\times X$. |
|
325 The proof that these two maps are inverse to each other is the same as in |
|
326 Theorem \ref{thm:product}. |
|
327 } |
|
328 |
|
329 |
322 |
330 \medskip |
323 \medskip |
331 |
324 |
332 \subsection{Reconstructing mapping spaces} |
325 \subsection{Reconstructing mapping spaces} |
333 \label{sec:map-recon} |
326 \label{sec:map-recon} |
359 \begin{proof} |
352 \begin{proof} |
360 The proof is again similar to that of Theorem \ref{thm:product}. |
353 The proof is again similar to that of Theorem \ref{thm:product}. |
361 |
354 |
362 We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
355 We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
363 |
356 |
364 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
357 Recall that |
365 $j$-fold mapping cylinders, $j \ge 0$. |
358 the 0-simplices of the homotopy colimit $\cB^\cT(M)$ |
366 So, as an abelian group (but not as a chain complex), |
359 are a direct sum of chain complexes with the summands indexed by |
367 \[ |
|
368 \cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
|
369 \] |
|
370 where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
|
371 |
|
372 Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
|
373 decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
360 decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
374 of $\cT$. |
361 of $\cT$. |
375 Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
362 Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
376 $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
363 $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
377 maps from the $n{-}1$-skeleton of $K$ to $T$. |
364 map from the $n{-}1$-skeleton of $K$ to $T$. |
378 The summand indexed by $(K, \vphi)$ is |
365 The summand indexed by $(K, \vphi)$ is |
379 \[ |
366 \[ |
380 \bigotimes_b D_*(b, \vphi), |
367 \bigotimes_b D_*(b, \vphi), |
381 \] |
368 \] |
382 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
369 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
383 chains of maps from $b$ to $T$ compatible with $\vphi$. |
370 chains of maps from $b$ to $T$ compatible with $\vphi$. |
384 We can take the product of these chains of maps to get a chains of maps from |
371 We can take the product of these chains of maps to get chains of maps from |
385 all of $M$ to $K$. |
372 all of $M$ to $K$. |
386 This defines $\psi$ on $C^0$. |
373 This defines $\psi$ on 0-simplices. |
387 |
374 |
388 We define $\psi(C^j) = 0$ for $j > 0$. |
375 We define $\psi$ to be zero on $(\ge1)$-simplices. |
389 It is not hard to see that this defines a chain map from |
376 It is not hard to see that this defines a chain map from |
390 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
377 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
391 |
378 |
392 The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by |
379 The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by |
393 families of maps whose support is contained in a disjoint union of balls. |
380 families of maps whose support is contained in a disjoint union of balls. |
405 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
392 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
406 Furthermore, we may choose $\phi$ such that for all $a$ |
393 Furthermore, we may choose $\phi$ such that for all $a$ |
407 \[ |
394 \[ |
408 \phi(a) = (a, K) + r |
395 \phi(a) = (a, K) + r |
409 \] |
396 \] |
410 where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$. |
397 where $(a, K)$ is a 0-simplex and $r$ is a sum of simplices of dimension 1 and greater. |
411 |
398 |
412 It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
399 It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
413 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
400 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
414 (See the proof of Theorem \ref{thm:product} for more details.) |
401 (See the proof of Theorem \ref{thm:product} for more details.) |
415 \end{proof} |
402 \end{proof} |
416 |
403 |
417 \noop{ |
|
418 % old proof (just start): |
|
419 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
|
420 We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology. |
|
421 |
|
422 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
|
423 $j$-fold mapping cylinders, $j \ge 0$. |
|
424 So, as an abelian group (but not as a chain complex), |
|
425 \[ |
|
426 \cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
|
427 \] |
|
428 where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
|
429 |
|
430 Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
|
431 decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
|
432 of $\cT$. |
|
433 Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
|
434 $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
|
435 maps from the $n{-}1$-skeleton of $K$ to $T$. |
|
436 The summand indexed by $(K, \vphi)$ is |
|
437 \[ |
|
438 \bigotimes_b D_*(b, \vphi), |
|
439 \] |
|
440 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
|
441 chains of maps from $b$ to $T$ compatible with $\vphi$. |
|
442 We can take the product of these chains of maps to get a chains of maps from |
|
443 all of $M$ to $K$. |
|
444 This defines $g$ on $C^0$. |
|
445 |
|
446 We define $g(C^j) = 0$ for $j > 0$. |
|
447 It is not hard to see that this defines a chain map from |
|
448 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
|
449 |
|
450 \nn{...} |
|
451 } |
|
452 |
|
453 \nn{maybe should also mention version where we enrich over |
404 \nn{maybe should also mention version where we enrich over |
454 spaces rather than chain complexes;} |
405 spaces rather than chain complexes;} |
455 |
406 |
456 \medskip |
407 \medskip |
457 \hrule |
408 \hrule |