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 |
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692 \nn{maybe also say that any splitting of $\bd c$ can be extended to a splitting of $c$} |
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693 |
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 |
699 We note two simple special cases of axiom \ref{axiom:vcones}. |
701 We note two simple special cases of Axiom \ref{axiom:vcones}. |
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 |
888 An $n$-category consists of the following data: |
890 An $n$-category consists of the following data: \nn{need to revise this list} |
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}); |