678 \medskip |
678 \medskip |
679 |
679 |
680 We need one additional axiom. |
680 We need one additional axiom. |
681 It says, roughly, that given a $k$-ball $X$, $k<n$, and $c\in \cC(X)$, there exist sufficiently many splittings of $c$. |
681 It says, roughly, that given a $k$-ball $X$, $k<n$, and $c\in \cC(X)$, there exist sufficiently many splittings of $c$. |
682 We use this axiom in the proofs of \ref{lem:d-a-acyclic} and \ref{lem:colim-injective}. |
682 We use this axiom in the proofs of \ref{lem:d-a-acyclic} and \ref{lem:colim-injective}. |
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683 The analogous axiom for systems of fields is used in the proof of \ref{small-blobs-b}. |
683 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
684 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
684 nevertheless we feel that it is too strong. |
685 nevertheless we feel that it is too strong. |
685 In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
686 In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
686 |
687 |
687 We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples. |
688 We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples. |
688 |
689 |
689 \begin{axiom}[Splittings] |
690 \begin{axiom}[Splittings] |
690 \label{axiom:splittings} |
691 \label{axiom:splittings} |
691 Let $c\in \cC_k(X)$, with $0\le k < n$. |
692 Let $c\in \cC_k(X)$, with $0\le k < n$. |
692 Let $X = \cup_i X_i$ be a splitting of $X$. |
693 Let $s = \{X_i\}$ be a splitting of X (so $X = \cup_i X_i$). |
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694 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which restrict to the identity on $\bd X$. |
693 \begin{itemize} |
695 \begin{itemize} |
694 \item (Version 1) Then there exists an embedded cell complex $S_c \sub X$, called the string locus of $c$, |
696 \item (Alternative 1) Consider the set of homeomorphisms $g:X\to X$ such that $c$ splits along $g(s)$. |
695 such that if the splitting $\{X_i\}$ is transverse to $S_c$ then $c$ splits along $\{X_i\}$. |
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696 \item (Version 2) Consider the set of homeomorphisms $g:X\to X$ such that $c$ splits along $\{g(X_i)\}$. |
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697 Then this subset of $\Homeo(X)$ is open and dense. |
697 Then this subset of $\Homeo(X)$ is open and dense. |
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698 Furthermore, if $s$ restricts to a splitting $\bd s$ of $\bd X$, and if $\bd c$ splits along $\bd s$, then the |
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699 intersection of the set of such homeomorphisms $g$ with $\Homeo_\bd(X)$ is open and dense in $\Homeo_\bd(X)$. |
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700 \item (Alternative 2) Then there exists an embedded cell complex $S_c \sub X$, called the string locus of $c$, |
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701 such that if the splitting $s$ is transverse to $S_c$ then $c$ splits along $s$. |
698 \end{itemize} |
702 \end{itemize} |
699 \nn{same something about extension from boundary} |
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700 \end{axiom} |
703 \end{axiom} |
701 |
704 |
702 We note some consequences of Axiom \ref{axiom:splittings}. |
705 We note some consequences of Axiom \ref{axiom:splittings}. |
703 |
706 |
704 First, some preliminary definitions. |
707 First, some preliminary definitions. |
825 (by Axiom \ref{axiom:splittings}) that $c$ splits along $q$. |
828 (by Axiom \ref{axiom:splittings}) that $c$ splits along $q$. |
826 We can now choose, for each splitting $p$ in $P$, a common refinement $p'$ of $p$ and $q$. |
829 We can now choose, for each splitting $p$ in $P$, a common refinement $p'$ of $p$ and $q$. |
827 This constitutes the middle part ($P\times \{0\}$ above) of $\vcone(P)$. |
830 This constitutes the middle part ($P\times \{0\}$ above) of $\vcone(P)$. |
828 \end{proof} |
831 \end{proof} |
829 |
832 |
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833 \begin{cor} |
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834 For any $c\in \cC_k(X)$, the geometric realization of the poset of splittings of $c$ is contractible. |
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835 \end{cor} |
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836 |
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837 \begin{proof} |
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838 In the geometric realization, V-Cones become actual cones, allowing us to contract any cycle. |
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839 \end{proof} |
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840 |
830 |
841 |
831 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
842 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
832 |
843 |
833 We need one additional axiom, in order to constrain the poset of decompositions of a given morphism. |
844 We need one additional axiom, in order to constrain the poset of decompositions of a given morphism. |
834 We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require |
845 We will soon want to take colimits (and homotopy colimits) indexed by such posets, and we want to require |
1751 \item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$; |
1762 \item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$; |
1752 \item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and |
1763 \item $b_i$ is the image of some $b'_i\in \psi(v'_i)$; and |
1753 \item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image |
1764 \item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image |
1754 of $b'_i$ and $b'_{i+1}$. |
1765 of $b'_i$ and $b'_{i+1}$. |
1755 \end{itemize} |
1766 \end{itemize} |
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1767 (This is possible by Axiom \ref{axiom:splittings}.) |
1756 Now consider the diagrams |
1768 Now consider the diagrams |
1757 \[ \xymatrix{ |
1769 \[ \xymatrix{ |
1758 & \psi(x'_{i-1}) \ar[rd] & \\ |
1770 & \psi(x'_{i-1}) \ar[rd] & \\ |
1759 \psi(v'_i) \ar[ru] \ar[rd] & & \psi(z) \\ |
1771 \psi(v'_i) \ar[ru] \ar[rd] & & \psi(z) \\ |
1760 & \psi(x'_i) \ar[ru] & |
1772 & \psi(x'_i) \ar[ru] & |