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736 When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category, |
736 When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category, |
737 which can be thought of as a free resolution. |
737 which can be thought of as a free resolution. |
738 \end{rem} |
738 \end{rem} |
739 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
739 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
740 |
740 |
741 %Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
741 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
742 %Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
742 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
743 The $A_\infty$ actions above allow us to state a gluing theorem. |
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744 For simplicity, we omit the $n$-category $\cC$ from the notation. |
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745 |
743 |
746 \begin{thm}[Gluing formula] |
744 \begin{thm}[Gluing formula] |
747 \label{thm:gluing} |
745 \label{thm:gluing} |
748 \mbox{}% <-- gets the indenting right |
746 \mbox{}% <-- gets the indenting right |
749 \begin{itemize} |
747 \begin{itemize} |
756 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
754 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
757 \end{equation*} |
755 \end{equation*} |
758 \end{itemize} |
756 \end{itemize} |
759 \end{thm} |
757 \end{thm} |
760 |
758 |
|
759 \begin{proof} (Sketch.) |
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760 |
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761 \end{proof} |
761 |
762 |
762 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
763 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
763 |
764 |
764 \begin{thm}[Product formula] |
765 \begin{thm}[Product formula] |
765 \label{thm:product} |
766 \label{thm:product} |