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1 %!TEX root = ../blob1.tex |
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2 |
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3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
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4 |
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5 \section{$n$-categories (maybe)} |
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6 \label{sec:ncats} |
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7 |
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8 \nn{experimental section. maybe this should be rolled into other sections. |
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9 maybe it should be split off into a separate paper.} |
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10 |
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11 Before proceeding, we need more appropriate definitions of $n$-categories, |
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12 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
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13 (As is the case throughout this paper, by ``$n$-category" we mean |
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14 a weak $n$-category with strong duality.) |
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15 |
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16 Consider first ordinary $n$-categories. |
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17 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
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18 We must decide on the ``shape" of the $k$-morphisms. |
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19 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
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20 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
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21 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
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22 and so on. |
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23 (This allows for strict associativity.) |
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24 Still other definitions \nn{need refs for all these; maybe the Leinster book} |
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25 model the $k$-morphisms on more complicated combinatorial polyhedra. |
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26 |
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27 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball. |
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28 In other words, |
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29 |
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30 \xxpar{Morphisms (preliminary version):}{For any $k$-manifold $X$ homeomorphic |
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31 to a $k$-ball, we have a set of $k$-morphisms |
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32 $\cC(X)$.} |
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33 |
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34 Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding |
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35 bijection of sets $f:\cC(X)\to \cC(Y)$. |
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36 So we replace the above with |
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37 |
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38 \xxpar{Morphisms:}{For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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39 the category of manifolds homeomorphic to the $k$-ball and |
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40 homeomorphisms to the category of sets and bijections.} |
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41 |
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42 (Note: We usually omit the subscript $k$.) |
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43 |
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44 We are being deliberately vague about what flavor of manifolds we are considering. |
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45 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
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46 They could be topological or PL or smooth. |
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47 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
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48 to be fussier about corners.) |
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49 For each flavor of manifold there is a corresponding flavor of $n$-category. |
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50 We will concentrate of the case of PL unoriented manifolds. |
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51 |
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52 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
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53 of morphisms). |
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54 The 0-sphere is unusual among spheres in that it is disconnected. |
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55 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
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56 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
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57 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
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58 We prefer to combine the domain and range into a single entity which we call the |
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59 boundary of a morphism. |
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60 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
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61 |
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62 \xxpar{Boundaries (domain and range), part 1:} |
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63 {For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
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64 the category of manifolds homeomorphic to the $k$-sphere and |
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65 homeomorphisms to the category of sets and bijections.} |
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66 |
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67 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
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68 |
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69 \xxpar{Boundaries, part 2:} |
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70 {For each $X$ homeomorphic to a $k$-ball, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
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71 These maps, for various $X$, comprise a natural transformation of functors.} |
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72 |
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73 (Note that the first ``$\bd$" above is part of the data for the category, |
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74 while the second is the ordinary boundary of manifolds.) |
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75 |
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76 Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$. |
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77 |
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78 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
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79 The various sets of $n$-morphisms $\cC(X; c)$, for all $X$ homeomorphic to an $n$-ball and |
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80 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
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81 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
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82 and all the structure maps of the $n$-category should be compatible with the auxiliary |
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83 category structure. |
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84 Note that this auxiliary structure is only in dimension $n$; |
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85 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
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86 |
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87 \medskip |
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88 \nn{At the moment I'm a little confused about orientations, and more specifically |
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89 about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
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90 Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
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91 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
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92 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
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93 equipped with an orientation of its once-stabilized tangent bundle. |
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94 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
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95 their $k$ times stabilized tangent bundles. |
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96 For the moment just stick with unoriented manifolds.} |
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97 \medskip |
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98 |
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99 We have just argued that the boundary of a morphism has no preferred splitting into |
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100 domain and range, but the converse meets with our approval. |
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101 That is, given compatible domain and range, we should be able to combine them into |
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102 the full boundary of a morphism: |
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103 |
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104 \xxpar{Domain $+$ range $\to$ boundary:} |
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105 {Let $S = B_1 \cup_E B_2$, where $S$ is homeomorphic to a $k$-sphere ($0\le k\le n-1$), |
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106 $B_i$ is homeomorphic to a $k$-ball, and $E = B_1\cap B_2$ is homeomorphic to a $k{-}1$-sphere. |
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107 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
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108 two maps $\bd: \cC(B_i)\to \cC(E)$. |
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109 Then (axiom) we have an injective map |
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110 \[ |
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111 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
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112 \] |
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113 which is natural with respect to the actions of homeomorphisms.} |
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114 |
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115 Note that we insist on injectivity above. |
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116 Let $\cC(S)_E$ denote the image of $\gl_E$. |
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117 We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as |
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118 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
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119 |
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120 If $B$ is homeomorphic to a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
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121 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
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122 |
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123 Next we consider composition of morphisms. |
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124 For $n$-categories which lack strong duality, one usually considers |
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125 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
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126 (For example, vertical and horizontal composition of 2-morphisms.) |
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127 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
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128 one general type of composition which can be in any ``direction". |
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129 |
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130 \xxpar{Composition:} |
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131 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are homeomorphic to $k$-balls ($0\le k\le n$) |
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132 and $Y = B_1\cap B_2$ is homeomorphic to a $k{-}1$-ball. |
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133 Let $E = \bd Y$, which is homeomorphic to a $k{-}2$-sphere. |
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134 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
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135 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
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136 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
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137 Then (axiom) we have a map |
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138 \[ |
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139 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
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140 \] |
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141 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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142 to the intersection of the boundaries of $B$ and $B_i$. |
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143 If $k < n$ we require that $\gl_Y$ is injective. |
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144 (For $k=n$, see below.)} |
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145 |
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146 |
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147 |