1684 this is a version of the familiar algebras-bimodules-intertwiners $2$-category. |
1684 this is a version of the familiar algebras-bimodules-intertwiners $2$-category. |
1685 It is clearly appropriate to call an $S^0$ module a bimodule, |
1685 It is clearly appropriate to call an $S^0$ module a bimodule, |
1686 but this is much less true for higher dimensional spheres, |
1686 but this is much less true for higher dimensional spheres, |
1687 so we prefer the term ``sphere module" for the general case. |
1687 so we prefer the term ``sphere module" for the general case. |
1688 |
1688 |
|
1689 The results of this subsection are not needed for the rest of the paper, |
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1690 so we will skimp on details in a couple of places. |
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1691 |
1689 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1692 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1690 |
1693 |
1691 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1694 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1692 these first. |
1695 these first. |
1693 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1696 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1694 of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
1697 of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
1695 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1698 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1696 the duality requirements of an $n{+}1$-category, we will have to assume |
1699 the duality requirements of an $n{+}1$-category, we will have to assume |
1985 z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
1988 z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
1986 \] |
1989 \] |
1987 In other words, the inner product on $Y$ is determined by the inner products on |
1990 In other words, the inner product on $Y$ is determined by the inner products on |
1988 $Y_1$, $Y_2$ and $D\times I$. |
1991 $Y_1$, $Y_2$ and $D\times I$. |
1989 |
1992 |
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1993 Now we show how to unambiguously identify $\cS(X; c; E)$ and $\cS(X; c; E')$ for any |
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1994 two choices of $E$ and $E'$. |
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1995 Consider first the case where $\bd X$ is decomposed as three $n$-balls $A$, $B$ and $C$, |
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1996 with $E = \bd(A\cup B)$ and $E' = \bd A$. |
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1997 We must provide an isomorphism between $\cS(X; c; E) = \hom(\cS(C), \cS(A\cup B))$ |
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1998 and $\cS(X; c; E') = \hom(\cS(C\cup \ol{B}), \cS(A))$. |
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1999 Let $D = B\cap A$. |
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2000 Then as above we can construct a map |
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2001 \[ |
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2002 \psi: \cS(B)\ot\cS(\ol{B}) \to \cS(D\times I) . |
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2003 \] |
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2004 Given $f\in \hom(\cS(C), \cS(A\cup B))$ we define $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ |
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2005 to be the composition |
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2006 \[ |
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2007 \cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
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2008 \cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
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2009 \cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
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2010 \] |
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2011 (See Figure xxxx.) |
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2012 Let $D' = B\cap C$. |
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2013 Using the inner products there is an adjoint map |
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2014 \[ |
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2015 \psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
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2016 \] |
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2017 Given $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ we define $f\in \hom(\cS(C), \cS(A\cup B))$ |
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2018 to be the composition |
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2019 \[ |
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2020 \cS(C) \stackrel{\cong}{\longrightarrow} |
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2021 \cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
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2022 \cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
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2023 \cS(A\cup B) . |
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2024 \] |
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2025 It is not hard too show that the above two maps are mutually inverse. |
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2026 |
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2027 \begin{lem} |
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2028 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
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2029 \end{lem} |
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2030 |
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2031 \begin{proof} |
|
2032 (Sketch) |
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2033 $E$ and $E'$ are isotopic, and any isotopy is |
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2034 homotopic to a composition of small isotopies which are either |
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2035 (a) supported away from $E$, or (b) modify $E$ in the simple manner described above. |
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2036 \end{proof} |
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2037 |
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2038 It follows from the lemma that we can construct an isomorphism |
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2039 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
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2040 This construction involves on a choice of simple ``moves" (as above) to transform |
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2041 $E$ to $E'$. |
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2042 We must now show that the isomorphism does not depend on this choice. |
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2043 We will show below that it suffice to check two ``movie moves". |
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2044 |
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2045 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
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2046 The result is equivalent to doing nothing. |
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2047 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
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2048 inverse, so we have invariance under this movie move. |
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2049 |
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2050 The second movie move replaces to successive pushes in the same direction, |
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2051 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
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2052 (See Figure xxxx.) |
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2053 Invariance under this movie move follows from the compatibility of the inner |
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2054 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
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2055 |
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2056 If $n\ge 2$, these two movie move suffice: |
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2057 |
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2058 \begin{lem} |
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2059 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
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2060 The any two sequences of elementary moves connecting $E$ to $E'$ |
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2061 are related by a sequence of the two movie moves defined above. |
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2062 \end{lem} |
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2063 |
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2064 \begin{proof} |
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2065 (Sketch) |
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2066 Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
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2067 of $\bd X$. |
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2068 Up to homotopy, |
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2069 such a family is homotopic to a family which can be decomposed |
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2070 into small families which are either |
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2071 (a) supported away from $E$, |
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2072 (b) have boundaries corresponding to the two movie moves above. |
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2073 Finally, observe that the space of $E$'s is simply connected. |
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2074 (This fails for $n=1$.) |
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2075 \end{proof} |
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2076 |
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2077 For $n=1$ we have to check an additional ``global" relations corresponding to |
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2078 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
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2079 \nn{should check this global move, or maybe cite Frobenius reciprocity result} |
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2080 |
1990 \nn{...} |
2081 \nn{...} |
1991 |
2082 |
1992 \medskip |
2083 \medskip |
1993 \hrule |
2084 \hrule |
1994 \medskip |
2085 \medskip |