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355 \begin{thm} \label{thm:map-recon} |
355 \begin{thm} \label{thm:map-recon} |
356 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
356 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
357 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
357 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
358 \end{thm} |
358 \end{thm} |
359 \begin{rem} |
359 \begin{rem} |
360 \nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...} |
360 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which is trivial at all but the topmost level. Ricardo Andrade also told us about a similar result. |
361 \nn{KW: Are you sure about that?} |
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362 Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level. |
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363 \end{rem} |
361 \end{rem} |
364 |
362 |
365 \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} |
363 \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} |
366 |
364 |
367 \begin{proof} |
365 \begin{proof} |