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233 |
233 |
234 Theorem \ref{product_thm} extends to the case of general fiber bundles |
234 Theorem \ref{product_thm} extends to the case of general fiber bundles |
235 \[ |
235 \[ |
236 F \to E \to Y . |
236 F \to E \to Y . |
237 \] |
237 \] |
238 We outline two approaches. |
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 \nn{need citation to other work that does this; Stolz and Teichner?} |
243 Call this a $k$-category over $Y$. |
243 Call this a $k$-category over $Y$. |
252 \bc_*(E) \simeq \cF_E(Y) . |
252 \bc_*(E) \simeq \cF_E(Y) . |
253 \] |
253 \] |
254 |
254 |
255 |
255 |
256 |
256 |
|
257 \nn{put this later} |
257 |
258 |
258 \nn{The second approach: Choose a decomposition $Y = \cup X_i$ |
259 \nn{The second approach: Choose a decomposition $Y = \cup X_i$ |
259 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
260 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
260 Choose the product structure as well. |
261 Choose the product structure as well. |
261 To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
262 To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
273 |
274 |
274 Next we prove a gluing theorem. |
275 Next we prove a gluing theorem. |
275 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 $X = X'_1\cup_Y X'_2$. |
276 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 |
277 $X = X_1\cup (Y\times J) \cup X_2$. |
278 $X = X_1\cup (Y\times J) \cup X_2$. |
278 \nn{need figure} |
|
279 Given this data we have: \nn{need refs to above for these} |
279 Given this data we have: \nn{need refs to above for these} |
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 $m+k = n$). \nn{need to explain $c$}. |