556 \end{axiom} |
556 \end{axiom} |
557 |
557 |
558 |
558 |
559 \medskip |
559 \medskip |
560 |
560 |
561 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
561 |
562 The last axiom (below), concerning actions of |
562 |
563 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
563 |
564 |
564 %All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
565 We start with the ordinary $n$-category case. |
565 %The last axiom (below), concerning actions of |
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566 %homeomorphisms in the top dimension $n$, distinguishes the two cases. |
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567 |
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568 %We start with the ordinary $n$-category case. |
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569 |
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570 The next axiom says, roughly, that we have strict associativity in dimension $n$, |
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571 even we we reparameterize our $n$-balls. |
566 |
572 |
567 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
573 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
568 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
574 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
569 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
575 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
570 Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$. |
576 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
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577 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
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578 trivially on $\bd b$. |
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579 Then $f(b) = b$. |
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580 In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on |
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581 all of $\cC(X)$. |
571 \end{axiom} |
582 \end{axiom} |
572 |
583 |
573 This axiom needs to be strengthened to force product morphisms to act as the identity. |
584 This axiom needs to be strengthened to force product morphisms to act as the identity. |
574 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
585 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
575 Let $J$ be a 1-ball (interval). |
586 Let $J$ be a 1-ball (interval). |
638 The revised axiom is |
649 The revised axiom is |
639 |
650 |
640 %\addtocounter{axiom}{-1} |
651 %\addtocounter{axiom}{-1} |
641 \begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$.] |
652 \begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$.] |
642 \label{axiom:extended-isotopies} |
653 \label{axiom:extended-isotopies} |
643 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
654 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
644 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
655 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
645 Then $f$ acts trivially on $\cC(X)$. |
656 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
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657 trivially on $\bd b$. |
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658 Then $f(b) = b$. |
646 In addition, collar maps act trivially on $\cC(X)$. |
659 In addition, collar maps act trivially on $\cC(X)$. |
647 \end{axiom} |
660 \end{axiom} |
648 |
661 |
649 \smallskip |
662 \medskip |
650 |
663 |
651 For $A_\infty$ $n$-categories, we replace |
664 |
652 isotopy invariance with the requirement that families of homeomorphisms act. |
665 |
653 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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654 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
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655 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
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656 |
666 |
657 \nn{begin temp relocation} |
667 \nn{begin temp relocation} |
658 |
668 |
659 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
669 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
660 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
670 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
670 \nn{$k=n$ injectivity for a-inf (necessary?)} |
680 \nn{$k=n$ injectivity for a-inf (necessary?)} |
671 or if $k=n$ and we are in the $A_\infty$ case, |
681 or if $k=n$ and we are in the $A_\infty$ case, |
672 |
682 |
673 |
683 |
674 \nn{end temp relocation} |
684 \nn{end temp relocation} |
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685 |
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686 |
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687 \smallskip |
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688 |
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689 For $A_\infty$ $n$-categories, we replace |
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690 isotopy invariance with the requirement that families of homeomorphisms act. |
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691 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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692 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
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693 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
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694 |
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695 |
675 |
696 |
676 %\addtocounter{axiom}{-1} |
697 %\addtocounter{axiom}{-1} |
677 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
698 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
678 \label{axiom:families} |
699 \label{axiom:families} |
679 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
700 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |