13 When we need to distinguish between the new and old definitions, we will refer to the 

    13 When we need to distinguish between the new and old definitions, we will refer to the 

    14 new-fangled and old-fashioned blob complex.

    14 new-fangled and old-fashioned blob complex.

    15 

    15 

    16 \medskip

    16 \medskip

    17 

    17 

    18 Let $M^n = Y^k\times F^{n-k}$.  

    24 Let $M^n = Y^k\times F^{n-k}$.  

    19 Let $C$ be a plain $n$-category.

    25 Let $C$ be a plain $n$-category.

    20 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball

    26 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball

    21 $X$ the old-fashioned blob complex $\bc_*(X\times F)$.

    27 $X$ the old-fashioned blob complex $\bc_*(X\times F)$.

    22 

    28 

    23 \begin{thm} \label{product_thm}

    29 \begin{thm} \label{product_thm}

    24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the

    30 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the

    25 new-fangled blob complex $\bc_*^\cF(Y)$.

    31 new-fangled blob complex $\bc_*^\cF(Y)$.

    26 \end{thm}

    32 \end{thm}

    27 

    33 

    29 

    35 

    30 \begin{proof}[Proof of Theorem \ref{product_thm}]

    36 \begin{proof}[Proof of Theorem \ref{product_thm}]

    31 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.

    37 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.

    32 

    38 

    33 First we define a map 

    39 First we define a map 

   211 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.

   217 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.

   212 \end{proof}

   218 \end{proof}

   213 

   219 

   214 \medskip

   220 \medskip

   215 

   221 

   216 Next we prove a gluing theorem.

   225 Next we prove a gluing theorem.

   217 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.

   226 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.

   218 We will need an explicit collar on $Y$, so rewrite this as

   227 We will need an explicit collar on $Y$, so rewrite this as

   219 $X = X_1\cup (Y\times J) \cup X_2$.

   228 $X = X_1\cup (Y\times J) \cup X_2$.

   220 \nn{need figure}

   229 \nn{need figure}

   228 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)

   237 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)

   229 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).

   238 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).

   230 \end{itemize}

   239 \end{itemize}

   231 

   240 

   232 \begin{thm}

   241 \begin{thm}

   233 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.

   243 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.

   234 \end{thm}

   244 \end{thm}

   235 

   245 

   236 \begin{proof}

   246 \begin{proof}

   237 The proof is similar to that of Theorem \ref{product_thm}.

   247 The proof is similar to that of Theorem \ref{product_thm}.

   251 \end{proof}

   261 \end{proof}

   252 

   262 

   253 This establishes Property \ref{property:gluing}.

   263 This establishes Property \ref{property:gluing}.

   254 

   264 

   255 \medskip

   265 \medskip

   256 

   268 

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

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

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

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

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

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

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

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