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3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
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5 \section{$n$-categories (maybe)} |
5 \section{$n$-categories} |
6 \label{sec:ncats} |
6 \label{sec:ncats} |
7 |
7 |
8 \nn{experimental section. maybe this should be rolled into other sections. |
8 %In order to make further progress establishing properties of the blob complex, |
9 maybe it should be split off into a separate paper.} |
9 %we need a definition of $A_\infty$ $n$-category that is adapted to our needs. |
10 |
10 %(Even in the case $n=1$, we need the new definition given below.) |
11 \nn{comment somewhere that what we really need is a convenient def of infty case, including tensor products etc. |
11 %It turns out that the $A_\infty$ $n$-category definition and the plain $n$-category |
12 but while we're at it might as well do plain case too.} |
12 %definition are mostly the same, so we give a new definition of plain |
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13 %$n$-categories too. |
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14 %We also define modules and tensor products for both plain and $A_\infty$ $n$-categories. |
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14 \subsection{Definition of $n$-categories} |
17 \subsection{Definition of $n$-categories} |
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18 |
16 Before proceeding, we need more appropriate definitions of $n$-categories, |
19 Before proceeding, we need more appropriate definitions of $n$-categories, |
17 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
20 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
18 (As is the case throughout this paper, by ``$n$-category" we mean |
21 (As is the case throughout this paper, by ``$n$-category" we mean |
19 a weak $n$-category with strong duality.) |
22 a weak $n$-category with strong duality.) |
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23 |
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24 The definitions presented below tie the categories more closely to the topology |
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25 and avoid combinatorial questions about, for example, the minimal sufficient |
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26 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
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27 For examples of topological origin, it is typically easy to show that they |
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28 satisfy our axioms. |
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29 For examples of a more purely algebraic origin, one would typically need the combinatorial |
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30 results that we have avoided here. |
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31 |
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32 \medskip |
20 |
33 |
21 Consider first ordinary $n$-categories. |
34 Consider first ordinary $n$-categories. |
22 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
35 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
23 We must decide on the ``shape" of the $k$-morphisms. |
36 We must decide on the ``shape" of the $k$-morphisms. |
24 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
37 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
50 bijection of sets $f:\cC(X)\to \cC(Y)$. |
63 bijection of sets $f:\cC(X)\to \cC(Y)$. |
51 (This will imply ``strong duality", among other things.) |
64 (This will imply ``strong duality", among other things.) |
52 So we replace the above with |
65 So we replace the above with |
53 |
66 |
54 \xxpar{Morphisms:} |
67 \xxpar{Morphisms:} |
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68 %\xxpar{Axiom 1 -- Morphisms:} |
55 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
69 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
56 the category of $k$-balls and |
70 the category of $k$-balls and |
57 homeomorphisms to the category of sets and bijections.} |
71 homeomorphisms to the category of sets and bijections.} |
58 |
72 |
59 (Note: We usually omit the subscript $k$.) |
73 (Note: We usually omit the subscript $k$.) |
114 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
128 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
115 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
129 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
116 equipped with an orientation of its once-stabilized tangent bundle. |
130 equipped with an orientation of its once-stabilized tangent bundle. |
117 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
131 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
118 their $k$ times stabilized tangent bundles. |
132 their $k$ times stabilized tangent bundles. |
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133 (cf. [Stolz and Teichner].) |
119 Probably should also have a framing of the stabilized dimensions in order to indicate which |
134 Probably should also have a framing of the stabilized dimensions in order to indicate which |
120 side the bounded manifold is on. |
135 side the bounded manifold is on. |
121 For the moment just stick with unoriented manifolds.} |
136 For the moment just stick with unoriented manifolds.} |
122 \medskip |
137 \medskip |
123 |
138 |