676 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
676 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
677 Here's a summary of the definition which segregates the data from the properties. |
677 Here's a summary of the definition which segregates the data from the properties. |
678 |
678 |
679 An $n$-category consists of the following data: |
679 An $n$-category consists of the following data: |
680 \begin{itemize} |
680 \begin{itemize} |
681 \item Functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}) |
681 \item Functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}). |
682 \item Boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}) |
682 \item Boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}). |
683 \item Composition/gluing maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}) |
683 \item Composition/gluing maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}). |
684 \item Product/identity maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}) |
684 \item Product/identity maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}). |
685 \item If enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ |
685 \item If enriching in an auxiliary category, additional structure on $\cC_n(X; c)$. |
686 \item In the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}) |
686 \item In the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
687 \end{itemize} |
687 \end{itemize} |
688 The above data must satisfy the following conditions: |
688 The above data must satisfy the following conditions: |
689 \begin{itemize} |
689 \begin{itemize} |
690 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
690 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
691 restrictions (Axiom \ref{axiom:composition}). |
691 restrictions (Axiom \ref{axiom:composition}). |