566 %homeomorphisms in the top dimension $n$, distinguishes the two cases. |
566 %homeomorphisms in the top dimension $n$, distinguishes the two cases. |
567 |
567 |
568 %We start with the ordinary $n$-category case. |
568 %We start with the ordinary $n$-category case. |
569 |
569 |
570 The next axiom says, roughly, that we have strict associativity in dimension $n$, |
570 The next axiom says, roughly, that we have strict associativity in dimension $n$, |
571 even we we reparameterize our $n$-balls. |
571 even when we reparametrize our $n$-balls. |
572 |
572 |
573 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
573 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
574 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
574 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
575 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
575 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
576 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
576 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
647 isotopic (rel boundary) to the identity {\it extended isotopy}. |
647 isotopic (rel boundary) to the identity {\it extended isotopy}. |
648 |
648 |
649 The revised axiom is |
649 The revised axiom is |
650 |
650 |
651 %\addtocounter{axiom}{-1} |
651 %\addtocounter{axiom}{-1} |
652 \begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$.] |
652 \begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$] |
653 \label{axiom:extended-isotopies} |
653 \label{axiom:extended-isotopies} |
654 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
654 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
655 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
655 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
656 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
656 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
657 act trivially on $\bd b$. |
657 act trivially on $\bd b$. |
658 Then $f(b) = b$. |
658 Then $f(b) = b$. |
659 In addition, collar maps act trivially on $\cC(X)$. |
659 In addition, collar maps act trivially on $\cC(X)$. |
660 \end{axiom} |
660 \end{axiom} |
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661 |
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662 \medskip |
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663 |
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664 We need one additional axiom, in order to constrain the poset of decompositions of a given morphism. |
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665 We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require |
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666 that these colimits are in some sense locally acyclic. |
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667 Before stating the axiom we need a few preliminary definitions. |
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668 If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$. |
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669 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$. |
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670 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone. |
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671 We call $P\times \{1\}$ the base of $\vcone(P)$. |
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672 (See Figure \nn{need figure}.) |
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673 |
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674 \nn{maybe call this ``splittings" instead of ``V-cones"?} |
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675 |
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676 \begin{axiom}[V-cones] |
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677 \label{axiom:vcones} |
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678 Let $c\in \cC_k(X)$ and |
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679 let $P$ be a finite poset of splittings of $c$. |
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680 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$. |
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681 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. |
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682 \end{axiom} |
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683 |
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684 It is easy to see that this axiom holds in our two motivating examples, |
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685 using standard facts about transversality and general position. |
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686 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams) |
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687 and also with respect to each of the decompositions of $P$, then chooses common refinements of each decomposition of $P$ |
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688 and the perturbed $q$. |
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689 These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$. |
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690 |
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691 We note two simple special cases of axiom \ref{axiom:vcones}. |
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692 If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$ |
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693 can be split along any decomposition of $X$, after a small perturbation. |
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694 If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the |
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695 poset of splittings of $c$ is connected. |
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696 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). |
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697 Two decompositions of $X$ might intersect in a very messy way, but one can always find a third |
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698 decomposition which has common refinements with each of the original two decompositions. |
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699 |
661 |
700 |
662 \medskip |
701 \medskip |
663 |
702 |
664 This completes the definition of an $n$-category. |
703 This completes the definition of an $n$-category. |
665 Next we define enriched $n$-categories. |
704 Next we define enriched $n$-categories. |