87 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
87 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
88 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
88 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
89 We prefer to combine the domain and range into a single entity which we call the |
89 We prefer to combine the domain and range into a single entity which we call the |
90 boundary of a morphism. |
90 boundary of a morphism. |
91 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
91 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
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92 |
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93 \nn{perhaps it's better to define $\cC(S^k)$ as a colimit, rather than making it new data} |
92 |
94 |
93 \begin{axiom}[Boundaries (spheres)] |
95 \begin{axiom}[Boundaries (spheres)] |
94 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
96 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
95 the category of $k$-spheres and |
97 the category of $k$-spheres and |
96 homeomorphisms to the category of sets and bijections. |
98 homeomorphisms to the category of sets and bijections. |
1078 |
1080 |
1079 |
1081 |
1080 |
1082 |
1081 \subsection{The $n{+}1$-category of sphere modules} |
1083 \subsection{The $n{+}1$-category of sphere modules} |
1082 |
1084 |
1083 In this subsection we define an $n{+}1$-category of ``sphere modules" whose objects |
1085 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1084 correspond to $n$-categories. |
1086 whose objects correspond to $n$-categories. |
1085 This is a version of the familiar algebras-bimodules-intertwinors 2-category. |
1087 This is a version of the familiar algebras-bimodules-intertwiners 2-category. |
1086 (Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule, |
1088 (Terminology: It is clearly appropriate to call an $S^0$ modules a bimodule, |
1087 since a 0-sphere has an obvious bi-ness. |
1089 since a 0-sphere has an obvious bi-ness. |
1088 This is much less true for higher dimensional spheres, |
1090 This is much less true for higher dimensional spheres, |
1089 so we prefer the term ``sphere module" for the general case.) |
1091 so we prefer the term ``sphere module" for the general case.) |
1090 |
1092 |
1091 |
1093 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1092 |
1094 these first. |
1093 \nn{need to assume a little extra structure to define the top ($n+1$) part (?)} |
1095 The $n{+}1$-dimensional part of $\cS$ consist of intertwiners |
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1096 (of garden-variety $1$-category modules associated to decorated $n$-balls). |
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1097 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
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1098 the duality requirements of an $n{+}1$-category, we will have to assume |
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1099 that our $n$-categories and modules have non-degenerate inner products. |
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1100 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
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1101 |
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1102 \medskip |
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1103 |
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1104 Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
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1105 These will be defined in terms of certain classes of marked balls, very similarly |
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1106 to the definition of $n$-category modules above. |
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1107 (This, in turn, is very similar to our definition of $n$-category.) |
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1108 Because of this similarity, we only sketch the definitions below. |
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1109 |
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1110 We start with 0-sphere modules, which also could reasonably be called (categorified) bimodules. |
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1111 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
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1112 Define a 0-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
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1113 $(B^k, B^{k-1})$, where $B^{k-1}$ is properly embedded in $B^k$. |
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1114 See Figure xxxx. |
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1115 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
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1116 |
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1117 0-marked balls can be cut into smaller balls in various ways. |
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1118 These smaller balls could be 0-marked or plain. |
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1119 We can also take the boundary of a 0-marked ball, which is 0-marked sphere. |
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1120 |
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1121 Fix $n$-categories $\cA$ and $\cB$. |
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1122 These will label the two halves of a 0-marked $k$-ball. |
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1123 The 0-sphere module we define next will depend on $\cA$ and $\cB$ |
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1124 (it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
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1125 |
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1126 An $n$-category 0-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
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1127 of 0-marked $k$-balls, $1\le k \le n$, |
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1128 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
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1129 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
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1130 Given a decomposition of a 0-marked $k$-ball $X$ into smaller balls $X_i$, we have |
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1131 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
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1132 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
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1133 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
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1134 Corresponding to this decomposition we have an action and/or composition map |
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1135 from the product of these various sets into $\cM(X)$. |
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1136 |
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1137 \medskip |
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1138 |
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1139 |
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1140 |
1094 |
1141 |
1095 \medskip |
1142 \medskip |
1096 \hrule |
1143 \hrule |
1097 \medskip |
1144 \medskip |
1098 |
1145 |