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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 \[ |
254 \gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)_E |
254 \gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B)\trans E |
255 \] |
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}); |
702 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}); |
702 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
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: |