21 For examples of a more purely algebraic origin, one would typically need the combinatorial |
21 For examples of a more purely algebraic origin, one would typically need the combinatorial |
22 results that we have avoided here. |
22 results that we have avoided here. |
23 |
23 |
24 \medskip |
24 \medskip |
25 |
25 |
26 Consider first ordinary $n$-categories. |
26 There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical chape. |
27 \nn{Actually, we're doing both plain and infinity cases here} |
27 Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). |
28 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$. |
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29 We must decide on the ``shape" of the $k$-morphisms. |
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30 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...). |
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31 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
28 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
32 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
29 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
33 and so on. |
30 and so on. |
34 (This allows for strict associativity.) |
31 (This allows for strict associativity.) |
35 Still other definitions \nn{need refs for all these; maybe the Leinster book \cite{MR2094071}} |
32 Still other definitions (see, for example, \cite{MR2094071}) |
36 model the $k$-morphisms on more complicated combinatorial polyhedra. |
33 model the $k$-morphisms on more complicated combinatorial polyhedra. |
37 |
34 |
38 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to |
35 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball: |
39 the standard $k$-ball. |
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40 In other words, |
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41 |
36 |
42 \begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
37 \begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
43 For any $k$-manifold $X$ homeomorphic |
38 For any $k$-manifold $X$ homeomorphic |
44 to the standard $k$-ball, we have a set of $k$-morphisms |
39 to the standard $k$-ball, we have a set of $k$-morphisms |
45 $\cC_k(X)$. |
40 $\cC_k(X)$. |
46 \end{preliminary-axiom} |
41 \end{preliminary-axiom} |
47 |
42 |
48 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
43 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
49 standard $k$-ball. |
44 standard $k$-ball. |
50 We {\it do not} assume that it is equipped with a |
45 We {\it do not} assume that it is equipped with a |
51 preferred homeomorphism to the standard $k$-ball. |
46 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
52 The same goes for ``a $k$-sphere" below. |
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53 |
47 |
54 |
48 |
55 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
49 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
56 the boundary), we want a corresponding |
50 the boundary), we want a corresponding |
57 bijection of sets $f:\cC(X)\to \cC(Y)$. |
51 bijection of sets $f:\cC(X)\to \cC(Y)$. |
82 of morphisms). |
76 of morphisms). |
83 The 0-sphere is unusual among spheres in that it is disconnected. |
77 The 0-sphere is unusual among spheres in that it is disconnected. |
84 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
78 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
85 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
79 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
86 by oriented 1-balls.) |
80 by oriented 1-balls.) |
87 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
81 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{A \tensor B^*}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place. |
88 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
82 |
89 We prefer to combine the domain and range into a single entity which we call the |
83 Instead, we combine the domain and range into a single entity which we call the |
90 boundary of a morphism. |
84 boundary of a morphism. |
91 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
85 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
92 |
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93 \nn{perhaps it's better to define $\cC(S^k)$ as a colimit, rather than making it new data} |
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94 |
86 |
95 \begin{axiom}[Boundaries (spheres)] |
87 \begin{axiom}[Boundaries (spheres)] |
96 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
88 For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
97 the category of $k$-spheres and |
89 the category of $k$-spheres and |
98 homeomorphisms to the category of sets and bijections. |
90 homeomorphisms to the category of sets and bijections. |
99 \end{axiom} |
91 \end{axiom} |
100 |
92 |
101 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
93 In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries, and again often omit the subscript. |
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94 |
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95 In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
102 |
96 |
103 \begin{axiom}[Boundaries (maps)]\label{nca-boundary} |
97 \begin{axiom}[Boundaries (maps)]\label{nca-boundary} |
104 For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
98 For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
105 These maps, for various $X$, comprise a natural transformation of functors. |
99 These maps, for various $X$, comprise a natural transformation of functors. |
106 \end{axiom} |
100 \end{axiom} |
107 |
101 |
108 (Note that the first ``$\bd$" above is part of the data for the category, |
102 (Note that the first ``$\bd$" above is part of the data for the category, |
109 while the second is the ordinary boundary of manifolds.) |
103 while the second is the ordinary boundary of manifolds.) |
110 |
104 |
111 Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
105 Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
112 |
106 |
113 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
107 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
114 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
108 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
115 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
109 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
116 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
110 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |