5 |
5 |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
7 a topological $n$-category from a traditional $n$-category; the morphisms of the |
7 a topological $n$-category from a traditional $n$-category; the morphisms of the |
8 topological $n$-category are string diagrams labeled by the traditional $n$-category. |
8 topological $n$-category are string diagrams labeled by the traditional $n$-category. |
9 In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
9 In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
10 The basic recipe, given a topological $n$-category $\cC$, is to define the $k$-morphisms |
10 The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms |
11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
12 $B^k$ is the {\it standard} $k$-ball. |
12 $B^k$ is the {\it standard} $k$-ball. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
14 One should also show that composing the two arrows (between traditional and topological $n$-categories) |
14 One should also show that composing the two arrows (between traditional and disk-like $n$-categories) |
15 yields the appropriate sort of equivalence on each side. |
15 yields the appropriate sort of equivalence on each side. |
16 Since we haven't given a definition for functors between topological $n$-categories |
16 Since we haven't given a definition for functors between disk-like $n$-categories |
17 (the paper is already too long!), we do not pursue this here. |
17 (the paper is already too long!), we do not pursue this here. |
18 |
18 |
19 We emphasize that we are just sketching some of the main ideas in this appendix --- |
19 We emphasize that we are just sketching some of the main ideas in this appendix --- |
20 it falls well short of proving the definitions are equivalent. |
20 it falls well short of proving the definitions are equivalent. |
21 |
21 |
22 %\nn{cases to cover: (a) ordinary $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
22 %\nn{cases to cover: (a) ordinary $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
23 %(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
23 %(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
24 |
24 |
25 \subsection{1-categories over \texorpdfstring{$\Set$ or $\Vect$}{Set or Vect}} |
25 \subsection{1-categories over \texorpdfstring{$\Set$ or $\Vect$}{Set or Vect}} |
26 \label{ssec:1-cats} |
26 \label{ssec:1-cats} |
27 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
27 Given a disk-like $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
28 This construction is quite straightforward, but we include the details for the sake of completeness, |
28 This construction is quite straightforward, but we include the details for the sake of completeness, |
29 because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
29 because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
30 on the underlying manifolds, and |
30 on the underlying manifolds, and |
31 to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
31 to shed some light on the $n=2$ case, which we describe in \S \ref{ssec:2-cats}. |
32 |
32 |
68 |
68 |
69 \noop{ |
69 \noop{ |
70 \medskip |
70 \medskip |
71 |
71 |
72 In the other direction, given a $1$-category $C$ |
72 In the other direction, given a $1$-category $C$ |
73 (with objects $C^0$ and morphisms $C^1$) we will construct a topological |
73 (with objects $C^0$ and morphisms $C^1$) we will construct a disk-like |
74 $1$-category $t(C)$. |
74 $1$-category $t(C)$. |
75 |
75 |
76 If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$. |
76 If $X$ is a 0-ball (point), let $t(C)(X) \deq C^0$. |
77 If $S$ is a 0-sphere, let $t(C)(S) \deq C^0\times C^0$. |
77 If $S$ is a 0-sphere, let $t(C)(S) \deq C^0\times C^0$. |
78 If $X$ is a 1-ball, let $t(C)(X) \deq C^1$. |
78 If $X$ is a 1-ball, let $t(C)(X) \deq C^1$. |
79 Homeomorphisms isotopic to the identity act trivially. |
79 Homeomorphisms isotopic to the identity act trivially. |
80 If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
80 If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure |
81 to define the action of homeomorphisms not isotopic to the identity |
81 to define the action of homeomorphisms not isotopic to the identity |
82 (and get, e.g., an unoriented topological 1-category). |
82 (and get, e.g., an unoriented disk-like 1-category). |
83 |
83 |
84 The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$. |
84 The domain and range maps of $C$ determine the boundary and restriction maps of $t(C)$. |
85 |
85 |
86 Gluing maps for $t(C)$ are determined by composition of morphisms in $C$. |
86 Gluing maps for $t(C)$ are determined by composition of morphisms in $C$. |
87 |
87 |
98 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
98 As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence. |
99 } %end \noop |
99 } %end \noop |
100 |
100 |
101 \medskip |
101 \medskip |
102 |
102 |
103 Similar arguments show that modules for topological 1-categories are essentially |
103 Similar arguments show that modules for disk-like 1-categories are essentially |
104 the same thing as traditional modules for traditional 1-categories. |
104 the same thing as traditional modules for traditional 1-categories. |
105 |
105 |
106 |
106 |
107 \subsection{Pivotal 2-categories} |
107 \subsection{Pivotal 2-categories} |
108 \label{ssec:2-cats} |
108 \label{ssec:2-cats} |
109 Let $\cC$ be a topological 2-category. |
109 Let $\cC$ be a disk-like 2-category. |
110 We will construct from $\cC$ a traditional pivotal 2-category. |
110 We will construct from $\cC$ a traditional pivotal 2-category. |
111 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
111 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
112 |
112 |
113 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
113 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
114 though this will make the $n=2$ case a little more complicated than necessary. |
114 though this will make the $n=2$ case a little more complicated than necessary. |
557 |
557 |
558 |
558 |
559 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
559 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
560 \label{sec:comparing-A-infty} |
560 \label{sec:comparing-A-infty} |
561 In this section, we make contact between the usual definition of an $A_\infty$ category |
561 In this section, we make contact between the usual definition of an $A_\infty$ category |
562 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
562 and our definition of a disk-like $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
563 |
563 |
564 \medskip |
564 \medskip |
565 |
565 |
566 Given a topological $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
566 Given a disk-like $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
567 $A_\infty$ $1$-category $A$ as follows. |
567 $A_\infty$ $1$-category $A$ as follows. |
568 The objects of $A$ are $\cC(pt)$. |
568 The objects of $A$ are $\cC(pt)$. |
569 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
569 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
570 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
570 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
571 For simplicity we will now assume there is only one object and suppress it from the notation. |
571 For simplicity we will now assume there is only one object and suppress it from the notation. |
603 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
603 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
604 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
604 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
605 Operad associativity for $A$ implies that this gluing map is independent of the choice of |
605 Operad associativity for $A$ implies that this gluing map is independent of the choice of |
606 $g$ and the choice of representative $(f_i, a_i)$. |
606 $g$ and the choice of representative $(f_i, a_i)$. |
607 |
607 |
608 It is straightforward to verify the remaining axioms for a topological $A_\infty$ 1-category. |
608 It is straightforward to verify the remaining axioms for a disk-like $A_\infty$ 1-category. |
609 |
609 |
610 |
610 |
611 |
611 |
612 |
612 |
613 |
613 |