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17 Compared to other definitions in the literature, |
17 Compared to other definitions in the literature, |
18 the definitions presented below tie the categories more closely to the topology |
18 the definitions presented below tie the categories more closely to the topology |
19 and avoid combinatorial questions about, for example, finding a minimal sufficient |
19 and avoid combinatorial questions about, for example, finding a minimal sufficient |
20 collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
20 collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
21 It is easy to show that examples of topological origin |
21 It is easy to show that examples of topological origin |
22 (e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories), |
22 (e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories) |
23 satisfy our axioms. |
23 satisfy our axioms. |
24 To show that examples of a more purely algebraic origin satisfy our axioms, |
24 To show that examples of a more purely algebraic origin satisfy our axioms, |
25 one would typically need the combinatorial |
25 one would typically need the combinatorial |
26 results that we have avoided here. |
26 results that we have avoided here. |
27 |
27 |
40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
41 |
41 |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
43 for $k{-}1$-morphisms. |
43 for $k{-}1$-morphisms. |
44 Readers who prefer things to be presented in a strictly logical order should read this |
44 Readers who prefer things to be presented in a strictly logical order should read this |
45 subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
45 subsection $n{+}1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
46 |
46 |
47 \medskip |
47 \medskip |
48 |
48 |
49 There are many existing definitions of $n$-categories, with various intended uses. |
49 There are many existing definitions of $n$-categories, with various intended uses. |
50 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
50 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
88 %\nn{need to check whether this makes much difference} |
88 %\nn{need to check whether this makes much difference} |
89 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
89 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
90 to be fussier about corners and boundaries.) |
90 to be fussier about corners and boundaries.) |
91 For each flavor of manifold there is a corresponding flavor of $n$-category. |
91 For each flavor of manifold there is a corresponding flavor of $n$-category. |
92 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
92 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
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93 |
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94 (An interesting open question is whether the techniques of this paper can be adapted to topological |
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95 manifolds and plain, merely continuous homeomorphisms. |
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96 The main obstacles are proving a version of Lemma \ref{basic_adaptation_lemma} and adapting the |
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97 transversality arguments used in Lemma \ref{lem:colim-injective}.) |
93 |
98 |
94 An ambitious reader may want to keep in mind two other classes of balls. |
99 An ambitious reader may want to keep in mind two other classes of balls. |
95 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
100 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
96 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
101 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
97 base space $Y$. |
102 base space $Y$. |
658 We call the equivalence relation generated by collar maps and homeomorphisms |
663 We call the equivalence relation generated by collar maps and homeomorphisms |
659 isotopic (rel boundary) to the identity {\it extended isotopy}. |
664 isotopic (rel boundary) to the identity {\it extended isotopy}. |
660 |
665 |
661 The revised axiom is |
666 The revised axiom is |
662 |
667 |
663 %\addtocounter{axiom}{-1} |
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664 \begin{axiom}[Extended isotopy invariance in dimension $n$] |
668 \begin{axiom}[Extended isotopy invariance in dimension $n$] |
665 \label{axiom:extended-isotopies} |
669 \label{axiom:extended-isotopies} |
666 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
670 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
667 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
671 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
668 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
672 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which |
673 |
677 |
674 \medskip |
678 \medskip |
675 |
679 |
676 We need one additional axiom. |
680 We need one additional axiom. |
677 It says, roughly, that given a $k$-ball $X$, $k<n$, and $c\in \cC(X)$, there exist sufficiently many splittings of $c$. |
681 It says, roughly, that given a $k$-ball $X$, $k<n$, and $c\in \cC(X)$, there exist sufficiently many splittings of $c$. |
678 We use this axiom in the proofs of \ref{lem:d-a-acyclic}, \ref{lem:colim-injective} \nn{...}. |
682 We use this axiom in the proofs of \ref{lem:d-a-acyclic} and \ref{lem:colim-injective}. |
679 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
683 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
680 nevertheless we feel that it is too strong. |
684 nevertheless we feel that it is too strong. |
681 In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
685 In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
682 |
686 |
683 We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples. |
687 We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples. |