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1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \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 $A_\infty$ $n$-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 following |
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. |
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8 \begin{defn} |
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9 The blob complex |
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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}. |
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11 \end{defn} |
8 |
12 |
9 We will show below |
13 We will show below |
10 in Corollary \ref{cor:new-old} |
14 in Corollary \ref{cor:new-old} |
11 that when $\cC$ is obtained from a system of fields $\cE$ |
15 that when $\cC$ is obtained from a system of fields $\cE$ |
12 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
16 as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), |
333 |
337 |
334 |
338 |
335 \subsection{A gluing theorem} |
339 \subsection{A gluing theorem} |
336 \label{sec:gluing} |
340 \label{sec:gluing} |
337 |
341 |
338 Next we prove a gluing theorem. |
342 Next we prove a gluing theorem. Throughout this section fix a particular $n$-dimensional system of fields $\cE$ and local relations. Each blob complex below is with respect to this $\cE$. |
339 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
343 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 |
344 We will need an explicit collar on $Y$, so rewrite this as |
341 $X = X_1\cup (Y\times J) \cup X_2$. |
345 $X = X_1\cup (Y\times J) \cup X_2$. |
342 Given this data we have: |
346 Given this data we have: |
343 \begin{itemize} |
347 \begin{itemize} |
362 (It will appear in a future paper.) |
366 (It will appear in a future paper.) |
363 So we content ourselves with |
367 So we content ourselves with |
364 |
368 |
365 \begin{thm} |
369 \begin{thm} |
366 \label{thm:gluing} |
370 \label{thm:gluing} |
367 When $k=n$ above, $\bc(X)$ is homotopy equivalent to $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
371 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 $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
368 \end{thm} |
372 \end{thm} |
369 |
373 |
370 \begin{proof} |
374 \begin{proof} |
371 %We will assume $k=n$; the other cases are similar. |
375 %We will assume $k=n$; the other cases are similar. |
372 The proof is similar to that of Theorem \ref{thm:product}. |
376 The proof is similar to that of Theorem \ref{thm:product}. |