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

   267 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.

   268 \end{proof}

   268 We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology.

   269 

   269 

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

   270 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of

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

   271 $j$-fold mapping cylinders, $j \ge 0$.

   272 So, as an abelian group (but not as a chain complex), 

   273 \[

   274 	\cB^\cT(M) = \bigoplus_{j\ge 0} C^j,

   275 \]

   276 where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders.

   277 

   278 Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by

   279 decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms

   280 of $\cT$.

   281 Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs

   282 $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous

   283 maps from the $n{-}1$-skeleton of $K$ to $T$.

   284 The summand indexed by $(K, \vphi)$ is

   285 \[

   286 	\bigotimes_b D_*(b, \vphi),

   287 \]

   288 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes

   289 chains of maps from $b$ to $T$ compatible with $\vphi$.

   290 We can take the product of these chains of maps to get a chains of maps from

   291 all of $M$ to $K$.

   292 This defines $g$ on $C^0$.

   293 

   294 We define $g(C^j) = 0$ for $j > 0$.

   295 It is not hard to see that this defines a chain map from 

   296 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.

   297 

   298 \nn{...}

   299 

   300 

   301 

   302 \end{proof}

   303 

   304 \nn{maybe should also mention version where we enrich over

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