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820 \end{example} |
820 \end{example} |
821 |
821 |
822 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
822 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
823 Notice that with $F$ a point, the above example is a construction turning a topological |
823 Notice that with $F$ a point, the above example is a construction turning a topological |
824 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
824 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
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825 \nn{do we use this notation elsewhere (anymore)?} |
825 We think of this as providing a ``free resolution" |
826 We think of this as providing a ``free resolution" |
826 of the topological $n$-category. |
827 of the topological $n$-category. |
827 \nn{say something about cofibrant replacements?} |
828 \nn{say something about cofibrant replacements?} |
828 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
829 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
829 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
830 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1411 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
1412 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
1412 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
1413 $\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
1413 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
1414 If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
1414 $\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
1415 $\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
1415 \end{example} |
1416 \end{example} |
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1417 |
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1418 \begin{example}[Examples from the blob complex] \label{bc-module-example} |
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1419 \rm |
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1420 In the previous example, we can instead define |
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1421 $\cF(Y)(M)\deq \bc_*^\cF((B\times W) \cup (N\times Y); c)$ (when $\dim(M) = n$) |
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1422 and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in |
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1423 Example \ref{ex:blob-complexes-of-balls}. |
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1424 \end{example} |
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1425 |
1416 |
1426 |
1417 \begin{example} |
1427 \begin{example} |
1418 \rm |
1428 \rm |
1419 Suppose $S$ is a topological space, with a subspace $T$. |
1429 Suppose $S$ is a topological space, with a subspace $T$. |
1420 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
1430 We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |