249 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.

   249 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.

   250 We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$.

   250 We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$.

   251 Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. 

   251 Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. 

   252 We have a map

   252 We have a map

   253 \[

   253 \[

   255 \]

   255 \]

   256 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions

   256 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions

   257 to the intersection of the boundaries of $B$ and $B_i$.

   257 to the intersection of the boundaries of $B$ and $B_i$.

   258 If $k < n$,

   258 If $k < n$,

   259 or if $k=n$ and we are in the $A_\infty$ case, 

   259 or if $k=n$ and we are in the $A_\infty$ case, 

   697 

   697 

   698 An $n$-category consists of the following data:

   698 An $n$-category consists of the following data:

   699 \begin{itemize}

   699 \begin{itemize}

   700 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});

   700 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});

   701 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});

   701 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});

   703 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product});

   703 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product});

   704 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$;

   704 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$;

   705 \item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}).

   705 \item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}).

   706 \end{itemize}

   706 \end{itemize}

   707 The above data must satisfy the following conditions:

   707 The above data must satisfy the following conditions: