430 where each $X_a$ is a $k$-ball. |
430 where each $X_a$ is a $k$-ball. |
431 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
431 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
432 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
432 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
433 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
433 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
434 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
434 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
435 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
435 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
436 \nn{need figures} |
436 See Figure \ref{partofJfig}.) |
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437 |
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438 \begin{figure}[!ht] |
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439 \begin{equation*} |
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440 \mathfig{.63}{tempkw/zz2} |
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441 \end{equation*} |
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442 \caption{A small part of $\cJ(W)$} |
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443 \label{partofJfig} |
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444 \end{figure} |
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445 |
437 |
446 |
438 $\cC$ determines |
447 $\cC$ determines |
439 a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
448 a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets |
440 (possibly with additional structure if $k=n$). |
449 (possibly with additional structure if $k=n$). |
441 For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
450 For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset |
602 In our example, the various restriction and gluing maps above come from |
611 In our example, the various restriction and gluing maps above come from |
603 restricting and gluing maps into $T$. |
612 restricting and gluing maps into $T$. |
604 |
613 |
605 We require two sorts of composition (gluing) for modules, corresponding to two ways |
614 We require two sorts of composition (gluing) for modules, corresponding to two ways |
606 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
615 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
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616 (See Figure \ref{zzz3}.) |
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617 |
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618 \begin{figure}[!ht] |
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619 \begin{equation*} |
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620 \mathfig{.63}{tempkw/zz3} |
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621 \end{equation*} |
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622 \caption{Module composition (top); $n$-category action (bottom)} |
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623 \label{zzz3} |
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624 \end{figure} |
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625 |
607 First, we can compose two module morphisms to get another module morphism. |
626 First, we can compose two module morphisms to get another module morphism. |
608 |
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609 \nn{need figures for next two axioms} |
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610 |
627 |
611 \xxpar{Module composition:} |
628 \xxpar{Module composition:} |
612 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
629 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
613 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
630 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
614 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
631 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
622 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
639 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
623 to the intersection of the boundaries of $M$ and $M_i$. |
640 to the intersection of the boundaries of $M$ and $M_i$. |
624 If $k < n$ we require that $\gl_Y$ is injective. |
641 If $k < n$ we require that $\gl_Y$ is injective. |
625 (For $k=n$, see below.)} |
642 (For $k=n$, see below.)} |
626 |
643 |
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644 |
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645 |
627 Second, we can compose an $n$-category morphism with a module morphism to get another |
646 Second, we can compose an $n$-category morphism with a module morphism to get another |
628 module morphism. |
647 module morphism. |
629 We'll call this the action map to distinguish it from the other kind of composition. |
648 We'll call this the action map to distinguish it from the other kind of composition. |
630 |
649 |
631 \xxpar{$n$-category action:} |
650 \xxpar{$n$-category action:} |
647 \xxpar{Module strict associativity:} |
666 \xxpar{Module strict associativity:} |
648 {The composition and action maps above are strictly associative.} |
667 {The composition and action maps above are strictly associative.} |
649 |
668 |
650 Note that the above associativity axiom applies to mixtures of module composition, |
669 Note that the above associativity axiom applies to mixtures of module composition, |
651 action maps and $n$-category composition. |
670 action maps and $n$-category composition. |
652 See Figure xxxx. |
671 See Figure \ref{zzz1b}. |
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672 |
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673 \begin{figure}[!ht] |
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674 \begin{equation*} |
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675 \mathfig{1}{tempkw/zz1b} |
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676 \end{equation*} |
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677 \caption{Two examples of mixed associativity} |
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678 \label{zzz1b} |
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679 \end{figure} |
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680 |
653 |
681 |
654 The above three axioms are equivalent to the following axiom, |
682 The above three axioms are equivalent to the following axiom, |
655 which we state in slightly vague form. |
683 which we state in slightly vague form. |
656 \nn{need figure for this} |
684 \nn{need figure for this} |
657 |
685 |
760 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
788 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
761 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
789 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
762 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
790 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
763 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
791 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
764 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
792 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
765 \nn{need figures} |
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766 |
793 |
767 $\cN$ determines |
794 $\cN$ determines |
768 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
795 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
769 (possibly with additional structure if $k=n$). |
796 (possibly with additional structure if $k=n$). |
770 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
797 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |