254 

   255 The next theorem shows how to reconstruct a mapping space from local data.

   256 Let $T$ be a topological space, let $M$ be an $n$-manifold, 

   257 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ 

   258 of Example \ref{ex:chains-of-maps-to-a-space}.

   259 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever

   260 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).

   261 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$.

   262 

   263 \begin{thm} \label{thm:map-recon}

   264 $\cB^\cT(M) \simeq C_*(\Maps(M\to T))$.

   265 \end{thm}

   266 \begin{proof}

   267 \nn{obvious map in one direction; use \ref{extension_lemma_b}; ...}

   268 \end{proof}

   269 

   270 \nn{should also mention version where we enrich over

   271 spaces rather than chain complexes; should comment on Lurie's (and others')  similar result

   272 for the $E_\infty$ case, and mention that our version does not require 

   273 any connectivity assumptions}

   274 

   275 \medskip