12 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
15 a ``weak" $n$-category with ``strong duality".) |
15 a ``weak" $n$-category with ``strong duality".) |
16 |
16 |
17 The definitions presented below tie the categories more closely to the topology |
17 Compared to other definitions in the literature, |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
18 the definitions presented below tie the categories more closely to the topology |
19 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
19 and avoid combinatorial questions about, for example, finding a minimal sufficient |
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20 collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
20 It is easy to show that examples of topological origin |
21 It is easy to show that examples of topological origin |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
22 (e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories), |
22 satisfy our axioms. |
23 satisfy our axioms. |
23 For examples of a more purely algebraic origin, one would typically need the combinatorial |
24 To show that examples of a more purely algebraic origin satisfy our axioms, |
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25 one would typically need the combinatorial |
24 results that we have avoided here. |
26 results that we have avoided here. |
25 |
27 |
26 See \S\ref{n-cat-names} for a discussion of $n$-category terminology. |
28 See \S\ref{n-cat-names} for a discussion of $n$-category terminology. |
27 |
29 |
28 %\nn{Say something explicit about Lurie's work here? |
30 %\nn{Say something explicit about Lurie's work here? |
29 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
31 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
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32 |
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33 \medskip |
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34 |
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35 The axioms for an $n$-category are spread throughout this section. |
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36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
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37 |
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38 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
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39 for $k{-}1$-morphisms. |
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40 So readers who prefer things to be presented in a strictly logical order should read this subsection $n$ times, first imagining that $k=0$, then that $k=1$, and so on until they reach $k=n$. |
30 |
41 |
31 \medskip |
42 \medskip |
32 |
43 |
33 There are many existing definitions of $n$-categories, with various intended uses. |
44 There are many existing definitions of $n$-categories, with various intended uses. |
34 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
45 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
47 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
58 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
48 standard $k$-ball. |
59 standard $k$-ball. |
49 We {\it do not} assume that it is equipped with a |
60 We {\it do not} assume that it is equipped with a |
50 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
61 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
51 |
62 |
52 The axioms for an $n$-category are spread throughout this section. |
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53 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
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54 |
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55 |
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56 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
63 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
57 the boundary), we want a corresponding |
64 the boundary), we want a corresponding |
58 bijection of sets $f:\cC(X)\to \cC(Y)$. |
65 bijection of sets $f:\cC_k(X)\to \cC_k(Y)$. |
59 (This will imply ``strong duality", among other things.) Putting these together, we have |
66 (This will imply ``strong duality", among other things.) Putting these together, we have |
60 |
67 |
61 \begin{axiom}[Morphisms] |
68 \begin{axiom}[Morphisms] |
62 \label{axiom:morphisms} |
69 \label{axiom:morphisms} |
63 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
70 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
101 Instead, we will combine the domain and range into a single entity which we call the |
108 Instead, we will combine the domain and range into a single entity which we call the |
102 boundary of a morphism. |
109 boundary of a morphism. |
103 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
110 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
104 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
111 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
105 $1\le k \le n$. |
112 $1\le k \le n$. |
106 At first it might seem that we need another axiom for this, but in fact once we have |
113 At first it might seem that we need another axiom |
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114 (more specifically, additional data) for this, but in fact once we have |
107 all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
115 all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
108 construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
116 construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
109 to spheres (and any other manifolds): |
117 to spheres (and any other manifolds): |
110 |
118 |
111 \begin{lem} |
119 \begin{lem} |
194 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
202 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
195 |
203 |
196 Note that we insist on injectivity above. |
204 Note that we insist on injectivity above. |
197 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
205 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
198 %\nn{we might want a more official looking proof...} |
206 %\nn{we might want a more official looking proof...} |
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207 |
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208 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
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209 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
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210 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
199 |
211 |
200 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
212 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
201 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
213 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
202 |
214 |
203 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
215 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |