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398 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
398 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
399 is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
399 is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
400 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
400 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
401 \end{thm} |
401 \end{thm} |
402 \begin{rem} |
402 \begin{rem} |
403 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology |
403 Lurie has shown in \cite[teorem 3.8.6]{0911.0018} that the topological chiral homology |
404 of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
404 of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
405 the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. |
405 the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. |
406 This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
406 This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
407 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
407 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
408 is trivial at levels 0 through $n-1$. |
408 is trivial at levels 0 through $n-1$. |