27 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball. |
27 We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to a $k$-ball. |
28 In other words, |
28 In other words, |
29 |
29 |
30 \xxpar{Morphisms (preliminary version):} |
30 \xxpar{Morphisms (preliminary version):} |
31 {For any $k$-manifold $X$ homeomorphic |
31 {For any $k$-manifold $X$ homeomorphic |
32 to a $k$-ball, we have a set of $k$-morphisms |
32 to the standard $k$-ball, we have a set of $k$-morphisms |
33 $\cC(X)$.} |
33 $\cC(X)$.} |
34 |
34 |
35 Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding |
35 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
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36 standard $k$-ball. |
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37 We {\it do not} assume that it is equipped with a |
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38 preferred homeomorphism to the standard $k$-ball. |
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39 The same goes for ``a $k$-sphere" below. |
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40 |
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41 Given a homeomorphism $f:X\to Y$ between $k$-balls, we want a corresponding |
36 bijection of sets $f:\cC(X)\to \cC(Y)$. |
42 bijection of sets $f:\cC(X)\to \cC(Y)$. |
37 So we replace the above with |
43 So we replace the above with |
38 |
44 |
39 \xxpar{Morphisms:} |
45 \xxpar{Morphisms:} |
40 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
46 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
41 the category of manifolds homeomorphic to the $k$-ball and |
47 the category of $k$-balls and |
42 homeomorphisms to the category of sets and bijections.} |
48 homeomorphisms to the category of sets and bijections.} |
43 |
49 |
44 (Note: We usually omit the subscript $k$.) |
50 (Note: We usually omit the subscript $k$.) |
45 |
51 |
46 We are being deliberately vague about what flavor of manifolds we are considering. |
52 We are being deliberately vague about what flavor of manifolds we are considering. |
53 |
59 |
54 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
60 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
55 of morphisms). |
61 of morphisms). |
56 The 0-sphere is unusual among spheres in that it is disconnected. |
62 The 0-sphere is unusual among spheres in that it is disconnected. |
57 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
63 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
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64 (Actually, this is only true in the oriented case.) |
58 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
65 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
59 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
66 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
60 We prefer to combine the domain and range into a single entity which we call the |
67 We prefer to combine the domain and range into a single entity which we call the |
61 boundary of a morphism. |
68 boundary of a morphism. |
62 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
69 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
63 |
70 |
64 \xxpar{Boundaries (domain and range), part 1:} |
71 \xxpar{Boundaries (domain and range), part 1:} |
65 {For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
72 {For each $0 \le k \le n-1$, we have a functor $\cC_k$ from |
66 the category of manifolds homeomorphic to the $k$-sphere and |
73 the category of $k$-spheres and |
67 homeomorphisms to the category of sets and bijections.} |
74 homeomorphisms to the category of sets and bijections.} |
68 |
75 |
69 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
76 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.) |
70 |
77 |
71 \xxpar{Boundaries, part 2:} |
78 \xxpar{Boundaries, part 2:} |
72 {For each $X$ homeomorphic to a $k$-ball, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
79 {For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$. |
73 These maps, for various $X$, comprise a natural transformation of functors.} |
80 These maps, for various $X$, comprise a natural transformation of functors.} |
74 |
81 |
75 (Note that the first ``$\bd$" above is part of the data for the category, |
82 (Note that the first ``$\bd$" above is part of the data for the category, |
76 while the second is the ordinary boundary of manifolds.) |
83 while the second is the ordinary boundary of manifolds.) |
77 |
84 |
78 Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$. |
85 Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$. |
79 |
86 |
80 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
87 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
81 The various sets of $n$-morphisms $\cC(X; c)$, for all $X$ homeomorphic to an $n$-ball and |
88 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
82 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
89 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
83 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
90 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
84 and all the structure maps of the $n$-category should be compatible with the auxiliary |
91 and all the structure maps of the $n$-category should be compatible with the auxiliary |
85 category structure. |
92 category structure. |
86 Note that this auxiliary structure is only in dimension $n$; |
93 Note that this auxiliary structure is only in dimension $n$; |
102 domain and range, but the converse meets with our approval. |
109 domain and range, but the converse meets with our approval. |
103 That is, given compatible domain and range, we should be able to combine them into |
110 That is, given compatible domain and range, we should be able to combine them into |
104 the full boundary of a morphism: |
111 the full boundary of a morphism: |
105 |
112 |
106 \xxpar{Domain $+$ range $\to$ boundary:} |
113 \xxpar{Domain $+$ range $\to$ boundary:} |
107 {Let $S = B_1 \cup_E B_2$, where $S$ is homeomorphic to a $k$-sphere ($0\le k\le n-1$), |
114 {Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$), |
108 $B_i$ is homeomorphic to a $k$-ball, and $E = B_1\cap B_2$ is homeomorphic to a $k{-}1$-sphere. |
115 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere. |
109 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
116 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the |
110 two maps $\bd: \cC(B_i)\to \cC(E)$. |
117 two maps $\bd: \cC(B_i)\to \cC(E)$. |
111 Then (axiom) we have an injective map |
118 Then (axiom) we have an injective map |
112 \[ |
119 \[ |
113 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
120 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
117 Note that we insist on injectivity above. |
124 Note that we insist on injectivity above. |
118 Let $\cC(S)_E$ denote the image of $\gl_E$. |
125 Let $\cC(S)_E$ denote the image of $\gl_E$. |
119 We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as |
126 We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as |
120 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
127 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
121 |
128 |
122 If $B$ is homeomorphic to a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
129 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
123 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
130 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
124 |
131 |
125 Next we consider composition of morphisms. |
132 Next we consider composition of morphisms. |
126 For $n$-categories which lack strong duality, one usually considers |
133 For $n$-categories which lack strong duality, one usually considers |
127 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
134 $k$ different types of composition of $k$-morphisms, each associated to a different direction. |
128 (For example, vertical and horizontal composition of 2-morphisms.) |
135 (For example, vertical and horizontal composition of 2-morphisms.) |
129 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
136 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
130 one general type of composition which can be in any ``direction". |
137 one general type of composition which can be in any ``direction". |
131 |
138 |
132 \xxpar{Composition:} |
139 \xxpar{Composition:} |
133 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are homeomorphic to $k$-balls ($0\le k\le n$) |
140 {Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
134 and $Y = B_1\cap B_2$ is homeomorphic to a $k{-}1$-ball. |
141 and $Y = B_1\cap B_2$ is a $k{-}1$-ball. |
135 Let $E = \bd Y$, which is homeomorphic to a $k{-}2$-sphere. |
142 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
136 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
143 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
137 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
144 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
138 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
145 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
139 Then (axiom) we have a map |
146 Then (axiom) we have a map |
140 \[ |
147 \[ |
248 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
255 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
249 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
256 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
250 \nn{need to also say something about collaring homeomorphisms.} |
257 \nn{need to also say something about collaring homeomorphisms.} |
251 \nn{this paragraph needs work.} |
258 \nn{this paragraph needs work.} |
252 |
259 |
253 Note that if take homology of chain complexes, we turn an $A_\infty$ $n$-category |
260 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
254 into a plain $n$-category. |
261 into a plain $n$-category (enriched over graded groups). |
255 \nn{say more here?} |
262 \nn{say more here?} |
256 In the other direction, if we enrich over topological spaces instead of chain complexes, |
263 In the other direction, if we enrich over topological spaces instead of chain complexes, |
257 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
264 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
258 instead of $C_*(\Homeo_\bd(X))$. |
265 instead of $C_*(\Homeo_\bd(X))$. |
259 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
266 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
287 \item Let $F$ be a closed $m$-manifold (e.g.\ a point). |
294 \item Let $F$ be a closed $m$-manifold (e.g.\ a point). |
288 Let $T$ be a topological space. |
295 Let $T$ be a topological space. |
289 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
296 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
290 all maps from $X\times F$ to $T$. |
297 all maps from $X\times F$ to $T$. |
291 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
298 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
292 homotopies fixed on $\bd X$. |
299 homotopies fixed on $\bd X \times F$. |
293 (Note that homotopy invariance implies isotopy invariance.) |
300 (Note that homotopy invariance implies isotopy invariance.) |
294 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
301 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
295 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
302 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
296 |
303 |
297 \item We can linearize the above example as follows. |
304 \item We can linearize the above example as follows. |
318 Define $\cC(X; c)$, for $X$ an $n$-ball, |
325 Define $\cC(X; c)$, for $X$ an $n$-ball, |
319 to be the dual Hilbert space $A(X\times F; c)$. |
326 to be the dual Hilbert space $A(X\times F; c)$. |
320 \nn{refer elsewhere for details?} |
327 \nn{refer elsewhere for details?} |
321 |
328 |
322 \item Variation on the above examples: |
329 \item Variation on the above examples: |
323 We could allow $F$ to have boundary and specify boundary conditions on $(\bd X)\times F$, |
330 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
324 for example product boundary conditions or take the union over all boundary conditions. |
331 for example product boundary conditions or take the union over all boundary conditions. |
325 \nn{maybe should not emphasize this case, since it's ``better" in some sense |
332 \nn{maybe should not emphasize this case, since it's ``better" in some sense |
326 to think of these guys as affording a representation |
333 to think of these guys as affording a representation |
327 of the $n{+}1$-category associated to $\bd F$.} |
334 of the $n{+}1$-category associated to $\bd F$.} |
328 |
335 |
371 (As with $n$-categories, we will usually omit the subscript $k$.) |
378 (As with $n$-categories, we will usually omit the subscript $k$.) |
372 |
379 |
373 In our example, let $\cM(B, N) = \cD((B\times \bd W)\cup_{N\times \bd W} (N\times W))$, |
380 In our example, let $\cM(B, N) = \cD((B\times \bd W)\cup_{N\times \bd W} (N\times W))$, |
374 where $\cD$ is the fields functor for the TQFT. |
381 where $\cD$ is the fields functor for the TQFT. |
375 |
382 |
376 Define the boundary of a marked ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
383 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
377 Call such a thing a {marked hemisphere}. |
384 Call such a thing a {marked $k{-}1$-hemisphere}. |
378 |
385 |
379 \xxpar{Module boundaries, part 1:} |
386 \xxpar{Module boundaries, part 1:} |
380 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
387 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
381 the category of marked hemispheres (of dimension $k$) and |
388 the category of marked hemispheres (of dimension $k$) and |
382 homeomorphisms to the category of sets and bijections.} |
389 homeomorphisms to the category of sets and bijections.} |
400 \[ |
407 \[ |
401 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
408 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
402 \] |
409 \] |
403 which is natural with respect to the actions of homeomorphisms.} |
410 which is natural with respect to the actions of homeomorphisms.} |
404 |
411 |
405 |
412 \xxpar{Axiom yet to be named:} |
406 |
413 {For each marked $k$-hemisphere $H$ there is a restriction map |
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414 $\cM(H)\to \cC(H)$. |
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415 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
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416 These maps comprise a natural transformation of functors.} |
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417 |
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418 Note that combining the various boundary and restriction maps above |
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419 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
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420 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
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421 This fact will be used below. |
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422 \nn{need to say more about splitableness/transversality in various places above} |
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423 |
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424 We stipulate two sorts of composition (gluing) for modules, corresponding to two ways |
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425 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
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426 First, we can compose two module morphisms to get another module morphism. |
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427 |
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428 \nn{need figures for next two axioms} |
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429 |
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430 \xxpar{Module composition:} |
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431 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$) |
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432 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
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433 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
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434 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
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435 We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
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436 Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
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437 Then (axiom) we have a map |
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438 \[ |
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439 \gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
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440 \] |
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441 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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442 to the intersection of the boundaries of $M$ and $M_i$. |
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443 If $k < n$ we require that $\gl_Y$ is injective. |
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444 (For $k=n$, see below.)} |
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445 |
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446 Second, we can compose an $n$-category morphism with a module morphism to get another |
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447 module morphism. |
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448 We'll call this the action map to distinguish it from the other kind of composition. |
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449 |
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450 \xxpar{$n$-category action:} |
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451 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
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452 $X$ is a plain $k$-ball, |
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453 and $Y = X\cap M'$ is a $k{-}1$-ball. |
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454 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
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455 We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
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456 Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
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457 Then (axiom) we have a map |
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458 \[ |
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459 \gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
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460 \] |
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461 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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462 to the intersection of the boundaries of $X$ and $M'$. |
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463 If $k < n$ we require that $\gl_Y$ is injective. |
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464 (For $k=n$, see below.)} |
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465 |
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466 \xxpar{Module strict associativity:} |
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467 {The composition and action maps above are strictly associative.} |
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468 |
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469 The above two axioms are equivalent to the following axiom, |
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470 which we state in slightly vague form. |
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471 \nn{need figure for this} |
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472 |
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473 \xxpar{Module multi-composition:} |
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474 {Given any decomposition |
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475 \[ |
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476 M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
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477 \] |
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478 of a marked $k$-ball $M$ |
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479 into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
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480 map from an appropriate subset (like a fibered product) |
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481 of |
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482 \[ |
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483 \cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
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484 \] |
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485 to $\cM(M)$, |
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486 and these various multifold composition maps satisfy an |
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487 operad-type strict associativity condition.} |
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488 |
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489 (The above operad-like structure is analogous to the swiss cheese operad |
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490 \nn{need citation}.) |
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491 \nn{need to double-check that this is true.} |
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492 |
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493 \xxpar{Module product (identity) morphisms:} |
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494 {Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$. |
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495 Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$. |
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496 If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram |
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497 \[ \xymatrix{ |
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498 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
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499 M \ar[r]^{f} & M' |
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500 } \] |
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501 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
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502 |
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503 \nn{Need to say something about compatibility with gluing (of both $M$ and $D$) above.} |
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504 |
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505 There are two alternatives for the next axiom, according whether we are defining |
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506 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
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507 In the plain case we require |
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508 |
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509 \xxpar{Pseudo and extended isotopy invariance in dimension $n$:} |
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510 {Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
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511 to the identity on $\bd M$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity. |
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512 Then $f$ acts trivially on $\cM(M)$.} |
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513 |
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514 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
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515 |
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516 We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
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517 In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
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518 on $\bd B \setmin N$. |
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519 |
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520 For $A_\infty$ modules we require |
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521 |
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522 \xxpar{Families of homeomorphisms act.} |
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523 {For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
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524 \[ |
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525 C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
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526 \] |
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527 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
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528 which fix $\bd M$. |
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529 These action maps are required to be associative up to homotopy |
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530 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
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531 a diagram like the one in Proposition \ref{CDprop} commutes. |
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532 \nn{repeat diagram here?} |
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533 \nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
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534 |
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535 \medskip |
407 |
536 |
408 |
537 |
409 |
538 |
410 |
539 |
411 \medskip |
540 \medskip |