1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} disk-like \texorpdfstring{$n$}{n}-categories} |
3 \section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories} |
4 \label{sec:ainfblob} |
4 \label{sec:ainfblob} |
5 Given an $A_\infty$ disk-like $n$-category $\cC$ and an $n$-manifold $M$, we make the |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the |
6 anticlimactically tautological definition of the blob |
6 anticlimactically tautological definition of the blob |
7 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
7 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
8 |
8 |
9 We will show below |
9 We will show below |
10 in Corollary \ref{cor:new-old} |
10 in Corollary \ref{cor:new-old} |
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 an $n$-dimensional 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 $A_\infty$ disk-like $k$-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 |
40 \begin{thm} \label{thm:product} |
40 \begin{thm} \label{thm:product} |
217 This concludes the proof of Theorem \ref{thm:product}. |
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 If $Y$ has dimension $k-m$, then we have a disk-like $m$-category $\cC_{Y\times F}$ whose value at |
222 If $Y$ has dimension $k-m$, then we have an $m$-category $\cC_{Y\times F}$ whose value at |
223 a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$ |
223 a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$ |
224 (if $j=m$). |
224 (if $j=m$). |
225 (See Example \ref{ex:blob-complexes-of-balls}.) |
225 (See Example \ref{ex:blob-complexes-of-balls}.) |
226 Similarly we have a disk-like $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$. |
226 Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$. |
227 These two categories are equivalent, but since we do not define functors between |
227 These two categories are equivalent, but since we do not define functors between |
228 disk-like $n$-categories in this paper we are unable to say precisely |
228 disk-like $n$-categories in this paper we are unable to say precisely |
229 what ``equivalent" means in this context. |
229 what ``equivalent" means in this context. |
230 We hope to include this stronger result in a future paper. |
230 We hope to include this stronger result in a future paper. |
231 |
231 |
233 |
233 |
234 Taking $F$ in Theorem \ref{thm:product} to be a point, we obtain the following corollary. |
234 Taking $F$ in Theorem \ref{thm:product} to be a point, we obtain the following corollary. |
235 |
235 |
236 \begin{cor} |
236 \begin{cor} |
237 \label{cor:new-old} |
237 \label{cor:new-old} |
238 Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ disk-like |
238 Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ |
239 $n$-category obtained from $\cE$ by taking the blob complex of balls. |
239 $n$-category obtained from $\cE$ by taking the blob complex of balls. |
240 Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are |
240 Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are |
241 homotopy equivalent: |
241 homotopy equivalent: |
242 \[ |
242 \[ |
243 \bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) . |
243 \bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) . |
259 calculation. |
259 calculation. |
260 |
260 |
261 We can generalize the definition of a $k$-category by replacing the categories |
261 We can generalize the definition of a $k$-category by replacing the categories |
262 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
262 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
263 (c.f. \cite{MR2079378}). |
263 (c.f. \cite{MR2079378}). |
264 Call this a disk-like $k$-category over $Y$. |
264 Call this a $k$-category over $Y$. |
265 A fiber bundle $F\to E\to Y$ gives an example of a disk-like $k$-category over $Y$: |
265 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
266 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
266 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
267 or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
267 or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
268 (Here $p^*(E)$ denotes the pull-back bundle over $D$.) |
268 (Here $p^*(E)$ denotes the pull-back bundle over $D$.) |
269 Let $\cF_E$ denote this disk-like $k$-category over $Y$. |
269 Let $\cF_E$ denote this $k$-category over $Y$. |
270 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
270 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
271 get a chain complex $\cl{\cF_E}(Y)$. |
271 get a chain complex $\cl{\cF_E}(Y)$. |
272 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
272 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
273 to show the following result. |
273 to show the following result. |
274 \begin{thm} |
274 \begin{thm} |
275 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the disk-like $k$-category over $Y$ defined above. |
275 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
276 Then |
276 Then |
277 \[ |
277 \[ |
278 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
278 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
279 \] |
279 \] |
280 \qed |
280 \qed |
285 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
285 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
286 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
286 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
287 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$. |
287 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$. |
288 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$ |
288 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$ |
289 lying above $D$.) |
289 lying above $D$.) |
290 We can define a disk-like $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$. |
290 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$. |
291 We can again adapt the homotopy colimit construction to |
291 We can again adapt the homotopy colimit construction to |
292 get a chain complex $\cl{\cF_M}(Y)$. |
292 get a chain complex $\cl{\cF_M}(Y)$. |
293 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
293 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
294 to show that |
294 to show that |
295 \begin{thm} |
295 \begin{thm} |
296 Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the disk-like $k$-category over $Y$ defined above. |
296 Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above. |
297 Then |
297 Then |
298 \[ |
298 \[ |
299 \bc_*(M) \simeq \cl{\cF_M}(Y) . |
299 \bc_*(M) \simeq \cl{\cF_M}(Y) . |
300 \] |
300 \] |
301 \qed |
301 \qed |
313 Let $F \to E \to Y$ be a fiber bundle as above. |
313 Let $F \to E \to Y$ be a fiber bundle as above. |
314 Choose a decomposition $Y = \cup X_i$ |
314 Choose a decomposition $Y = \cup X_i$ |
315 such that the restriction of $E$ to $X_i$ is homeomorphic to a product $F\times X_i$, |
315 such that the restriction of $E$ to $X_i$ is homeomorphic to a product $F\times X_i$, |
316 and choose trivializations of these products as well. |
316 and choose trivializations of these products as well. |
317 |
317 |
318 Let $\cF$ be the disk-like $k$-category associated to $F$. |
318 Let $\cF$ be the $k$-category associated to $F$. |
319 To each codimension-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module) for $\cF$. |
319 To each codimension-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module) for $\cF$. |
320 More generally, to each codimension-$m$ face we have an $S^{m-1}$-module for a $(k{-}m{+}1)$-category |
320 More generally, to each codimension-$m$ face we have an $S^{m-1}$-module for a $(k{-}m{+}1)$-category |
321 associated to the (decorated) link of that face. |
321 associated to the (decorated) link of that face. |
322 We can decorate the strata of the decomposition of $Y$ with these sphere modules and form a |
322 We can decorate the strata of the decomposition of $Y$ with these sphere modules and form a |
323 colimit as in \S \ref{ssec:spherecat}. |
323 colimit as in \S \ref{ssec:spherecat}. |
339 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
339 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
340 We will need an explicit collar on $Y$, so rewrite this as |
340 We will need an explicit collar on $Y$, so rewrite this as |
341 $X = X_1\cup (Y\times J) \cup X_2$. |
341 $X = X_1\cup (Y\times J) \cup X_2$. |
342 Given this data we have: |
342 Given this data we have: |
343 \begin{itemize} |
343 \begin{itemize} |
344 \item An $A_\infty$ disk-like $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
344 \item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
345 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
345 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
346 (for $m+k = n$). |
346 (for $m+k = n$). |
347 (See Example \ref{ex:blob-complexes-of-balls}.) |
347 (See Example \ref{ex:blob-complexes-of-balls}.) |
348 %\nn{need to explain $c$}. |
348 %\nn{need to explain $c$}. |
349 \item An $A_\infty$ disk-like $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
349 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
350 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
350 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
351 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
351 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
352 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
352 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
353 (See Example \ref{bc-module-example}.) |
353 (See Example \ref{bc-module-example}.) |
354 \item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is |
354 \item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is |
355 an $A_\infty$ disk-like $n{-}k$-category. |
355 an $A_\infty$ $n{-}k$-category. |
356 (See \S \ref{moddecss}.) |
356 (See \S \ref{moddecss}.) |
357 \end{itemize} |
357 \end{itemize} |
358 |
358 |
359 It is the case that the disk-like $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$ |
359 It is the case that the $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$ |
360 are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories |
360 are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories |
361 in this paper, we cannot state this precisely. |
361 in this paper, we cannot state this precisely. |
362 (It will appear in a future paper.) |
362 (It will appear in a future paper.) |
363 So we content ourselves with |
363 So we content ourselves with |
364 |
364 |
401 \subsection{Reconstructing mapping spaces} |
401 \subsection{Reconstructing mapping spaces} |
402 \label{sec:map-recon} |
402 \label{sec:map-recon} |
403 |
403 |
404 The next theorem shows how to reconstruct a mapping space from local data. |
404 The next theorem shows how to reconstruct a mapping space from local data. |
405 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
405 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
406 and recall the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$ |
406 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
407 of Example \ref{ex:chains-of-maps-to-a-space}. |
407 of Example \ref{ex:chains-of-maps-to-a-space}. |
408 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
408 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
409 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
409 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
410 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
410 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
411 |
411 |