687 let $P$ be a finite poset of splittings of $c$.

   687 let $P$ be a finite poset of splittings of $c$.

   688 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   688 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   689 Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.

   689 Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.

   690 \end{axiom}

   690 \end{axiom}

   691 

   691 

   692 It is easy to see that this axiom holds in our two motivating examples, 

   694 It is easy to see that this axiom holds in our two motivating examples, 

   693 using standard facts about transversality and general position.

   695 using standard facts about transversality and general position.

   694 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams)

   696 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams)

   695 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$

   697 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$

   696 and the perturbed $q$.

   698 and the perturbed $q$.

   697 These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$.

   699 These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$.

   698 

   700 

   700 If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$

   702 If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$

   701 can be split along any decomposition of $X$, after a small perturbation.

   703 can be split along any decomposition of $X$, after a small perturbation.

   702 If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the

   704 If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the

   703 poset of splittings of $c$ is connected.

   705 poset of splittings of $c$ is connected.

   704 Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V).

   706 Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V).

   883 a colimit construction; see \S \ref{ss:ncat_fields} below.

   885 a colimit construction; see \S \ref{ss:ncat_fields} below.

   884 

   886 

   885 In the $n$-category axioms above we have intermingled data and properties for expository reasons.

   887 In the $n$-category axioms above we have intermingled data and properties for expository reasons.

   886 Here's a summary of the definition which segregates the data from the properties.

   888 Here's a summary of the definition which segregates the data from the properties.

   887 

   889 

   889 \begin{itemize}

   891 \begin{itemize}

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

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

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

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

   892 \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});

   894 \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});

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

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