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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Comparing $n$-category definitions} |
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4 \label{sec:comparing-defs} |
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5 |
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6 In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats} |
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7 to more traditional definitions, for $n=1$ and 2. |
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8 |
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9 \subsection{Plain 1-categories} |
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10 |
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11 Given a topological 1-category $\cC$, we construct a traditional 1-category $C$. |
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12 (This is quite straightforward, but we include the details for the sake of completeness and |
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13 to shed some light on the $n=2$ case.) |
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14 |
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15 Let the objects of $C$ be $C^0 \deq \cC(B^0)$ and the morphisms of $C$ be $C^1 \deq \cC(B^1)$, |
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16 where $B^k$ denotes the standard $k$-ball. |
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17 The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$. |
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18 |
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19 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$. |
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20 Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree). |
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21 By isotopy invariance in $C$, any other choice of homeomorphism gives the same composition rule. |
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22 |
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23 Given $a\in C^0$, define $\id_a \deq a\times B^1$. |
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24 By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism. |
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25 |
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26 \nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?} |
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27 |
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28 \medskip |
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29 |
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30 For 1-categories based on oriented manifolds, there is no additional structure. |
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31 |
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32 For 1-categories based on unoriented manifolds, there is a map $*:C^1\to C^1$ |
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33 coming from $\cC$ applied to an orientation-reversing homeomorphism (unique up to isotopy) |
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34 from $B^1$ to itself. |
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35 Topological properties of this homeomorphism imply that |
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36 $a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$ |
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37 (* is an anti-automorphism). |
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38 |
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39 For 1-categories based on Spin manifolds, |
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40 the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity |
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41 gives an order 2 automorphism of $C^1$. |
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42 |
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43 For 1-categories based on $\text{Pin}_-$ manifolds, |
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44 we have an order 4 antiautomorphism of $C^1$. |
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45 |
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46 For 1-categories based on $\text{Pin}_+$ manifolds, |
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47 we have an order 2 antiautomorphism and also an order 2 automorphism of $C^1$, |
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48 and these two maps commute with each other. |
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49 |
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50 \nn{need to also consider automorphisms of $B^0$ / objects} |
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51 |
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52 \medskip |
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53 |
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54 In the other direction, given a traditional 1-category $C$ |
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55 (with objects $C^0$ and morphisms $C^1$) we will construct a topological |
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56 1-category $\cC$. |
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57 |
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58 If $X$ is a 0-ball (point), let $\cC(X) \deq C^0$. |
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59 If $S$ is a 0-sphere, let $\cC(S) \deq C^0\times C^0$. |
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60 If $X$ is a 1-ball, let $\cC(X) \deq C^1$. |
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61 Homeomorphisms isotopic to the identity act trivially. |
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62 If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
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63 to define the action of homeomorphisms not isotopic to the identity |
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64 (and get, e.g., an unoriented topological 1-category). |
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65 |
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66 The domain and range maps of $C$ determine the boundary and restriction maps of $\cC$. |
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67 |
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68 Gluing maps for $\cC$ are determined my composition of morphisms in $C$. |
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69 |
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70 For $X$ a 0-ball, $D$ a 1-ball and $a\in \cC(X)$, define the product morphism |
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71 $a\times D \deq \id_a$. |
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72 It is not hard to verify that this has the desired properties. |
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73 |
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74 \medskip |
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75 |
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76 The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back |
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77 more or less exactly the same thing we started with. |
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78 \nn{need better notation here} |
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79 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
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80 |
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81 |
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82 \subsection{Plain 2-categories} |
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83 |
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84 blah |
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85 \nn{...} |
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86 |
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87 \medskip |
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88 \hrule |
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89 \medskip |
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90 |
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91 \nn{to be continued...} |
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92 \medskip |