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4 \label{sec:ainfblob} |
4 \label{sec:ainfblob} |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the following |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the following |
6 anticlimactically tautological definition of the blob |
6 anticlimactically tautological definition of the blob |
7 complex. |
7 complex. |
8 \begin{defn} |
8 \begin{defn} |
9 The blob complex |
9 The blob complex $\bc_*(M;\cC)$ of an $n$-manifold $M$ with coefficients in |
10 $\bc_*(M;\cC)$ of an $n$-manifold $n$ with coefficients in an $A_\infty$ $n$-category is the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
10 an $A_\infty$ $n$-category $\cC$ is the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
11 \end{defn} |
11 \end{defn} |
12 |
12 |
13 We will show below |
13 We will show below |
14 in Corollary \ref{cor:new-old} |
14 in Corollary \ref{cor:new-old} |
15 that when $\cC$ is obtained from a system of fields $\cE$ |
15 that when $\cC$ is obtained from a system of fields $\cE$ |
385 (It will appear in a future paper.) |
385 (It will appear in a future paper.) |
386 So we content ourselves with |
386 So we content ourselves with |
387 |
387 |
388 \begin{thm} |
388 \begin{thm} |
389 \label{thm:gluing} |
389 \label{thm:gluing} |
390 Suppose $X$ is an $n$-manifold, and $X = X_1\cup (Y\times J) \cup X_2$ (i.e. just as with $k=n$ above). Then $\bc(X)$ is homotopy equivalent to the $A_\infty$ tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
390 Suppose $X$ is an $n$-manifold, and $X = X_1\cup (Y\times J) \cup X_2$ (i.e. take $k=n$ in the above discussion). |
|
391 Then $\bc(X)$ is homotopy equivalent to the $A_\infty$ tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
391 \end{thm} |
392 \end{thm} |
392 |
393 |
393 \begin{proof} |
394 \begin{proof} |
394 %We will assume $k=n$; the other cases are similar. |
395 %We will assume $k=n$; the other cases are similar. |
395 The proof is similar to that of Theorem \ref{thm:product}. |
396 The proof is similar to that of Theorem \ref{thm:product}. |
413 As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
414 As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
414 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
415 an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
415 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
416 The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
416 a common refinement. |
417 a common refinement. |
417 |
418 |
418 The proof that these two maps are inverse to each other is the same as in |
419 The proof that these two maps are homotopy inverse to each other is the same as in |
419 Theorem \ref{thm:product}. |
420 Theorem \ref{thm:product}. |
420 \end{proof} |
421 \end{proof} |
421 |
422 |
422 \medskip |
423 \medskip |
423 |
424 |