1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
2 |
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3 \section{Comparing $n$-category definitions} |
3 \section{Comparing $n$-category definitions} |
4 \label{sec:comparing-defs} |
4 \label{sec:comparing-defs} |
5 |
5 |
6 In this appendix we relate the ``topological" category definitions of \S\ref{sec:ncats} |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
7 to more traditional definitions, for $n=1$ and 2. |
7 a topological $n$-category from a traditional $n$-category; the morphisms of the |
8 |
8 topological $n$-category are string diagrams labeled by the traditional $n$-category. |
9 \nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
9 In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
10 (c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
10 The basic recipe, given a topological $n$-category $\cC$, is to define the $k$-morphisms |
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11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
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12 $B^k$ is the {\it standard} $k$-ball. |
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13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
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14 One should also show that composing the two arrows (between traditional and topological $n$-categories) |
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15 yields the appropriate sort of equivalence on each side. |
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16 Since we haven't given a definition for functors between topological $n$-categories |
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17 (the paper is already too long!), we do not pursue this here. |
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18 \nn{say something about modules and tensor products?} |
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19 |
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20 We emphasize that we are just sketching some of the main ideas in this appendix --- |
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21 it falls well short of proving the definitions are equivalent. |
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22 |
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23 %\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
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24 %(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
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25 |
12 \subsection{$1$-categories over $\Set$ or $\Vect$} |
26 \subsection{$1$-categories over $\Set$ or $\Vect$} |
13 \label{ssec:1-cats} |
27 \label{ssec:1-cats} |
14 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
28 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
15 This construction is quite straightforward, but we include the details for the sake of completeness, |
29 This construction is quite straightforward, but we include the details for the sake of completeness, |
32 |
46 |
33 |
47 |
34 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
48 If the underlying manifolds for $\cX$ have further geometric structure, then we obtain certain functors. |
35 The base case is for oriented manifolds, where we obtain no extra algebraic data. |
49 The base case is for oriented manifolds, where we obtain no extra algebraic data. |
36 |
50 |
37 For 1-categories based on unoriented manifolds (somewhat confusingly, we're thinking of being |
51 For 1-categories based on unoriented manifolds, |
38 unoriented as requiring extra data beyond being oriented, namely the identification between the orientations), |
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39 there is a map $*:c(\cX)^1\to c(\cX)^1$ |
52 there is a map $*:c(\cX)^1\to c(\cX)^1$ |
40 coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
53 coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
41 from $B^1$ to itself. |
54 from $B^1$ to itself. |
42 Topological properties of this homeomorphism imply that |
55 Topological properties of this homeomorphism imply that |
43 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
56 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
50 For 1-categories based on $\text{Pin}_-$ manifolds, |
63 For 1-categories based on $\text{Pin}_-$ manifolds, |
51 we have an order 4 antiautomorphism of $c(\cX)^1$. |
64 we have an order 4 antiautomorphism of $c(\cX)^1$. |
52 For 1-categories based on $\text{Pin}_+$ manifolds, |
65 For 1-categories based on $\text{Pin}_+$ manifolds, |
53 we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
66 we have an order 2 antiautomorphism and also an order 2 automorphism of $c(\cX)^1$, |
54 and these two maps commute with each other. |
67 and these two maps commute with each other. |
55 \nn{need to also consider automorphisms of $B^0$ / objects} |
68 %\nn{need to also consider automorphisms of $B^0$ / objects} |
56 |
69 |
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70 \noop{ |
57 \medskip |
71 \medskip |
58 |
72 |
59 In the other direction, given a $1$-category $C$ |
73 In the other direction, given a $1$-category $C$ |
60 (with objects $C^0$ and morphisms $C^1$) we will construct a topological |
74 (with objects $C^0$ and morphisms $C^1$) we will construct a topological |
61 $1$-category $t(C)$. |
75 $1$-category $t(C)$. |
81 The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ |
95 The compositions of the constructions above, $$\cX\to c(\cX)\to t(c(\cX))$$ |
82 and $$C\to t(C)\to c(t(C)),$$ give back |
96 and $$C\to t(C)\to c(t(C)),$$ give back |
83 more or less exactly the same thing we started with. |
97 more or less exactly the same thing we started with. |
84 |
98 |
85 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
99 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
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100 } %end \noop |
86 |
101 |
87 \medskip |
102 \medskip |
88 |
103 |
89 Similar arguments show that modules for topological 1-categories are essentially |
104 Similar arguments show that modules for topological 1-categories are essentially |
90 the same thing as traditional modules for traditional 1-categories. |
105 the same thing as traditional modules for traditional 1-categories. |
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106 |
91 |
107 |
92 \subsection{Plain 2-categories} |
108 \subsection{Plain 2-categories} |
93 \label{ssec:2-cats} |
109 \label{ssec:2-cats} |
94 Let $\cC$ be a topological 2-category. |
110 Let $\cC$ be a topological 2-category. |
95 We will construct a traditional pivotal 2-category. |
111 We will construct a traditional pivotal 2-category. |