31 In other words, |
31 In other words, |
32 |
32 |
33 \xxpar{Morphisms (preliminary version):} |
33 \xxpar{Morphisms (preliminary version):} |
34 {For any $k$-manifold $X$ homeomorphic |
34 {For any $k$-manifold $X$ homeomorphic |
35 to the standard $k$-ball, we have a set of $k$-morphisms |
35 to the standard $k$-ball, we have a set of $k$-morphisms |
36 $\cC(X)$.} |
36 $\cC_k(X)$.} |
37 |
37 |
38 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
38 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
39 standard $k$-ball. |
39 standard $k$-ball. |
40 We {\it do not} assume that it is equipped with a |
40 We {\it do not} assume that it is equipped with a |
41 preferred homeomorphism to the standard $k$-ball. |
41 preferred homeomorphism to the standard $k$-ball. |
42 The same goes for ``a $k$-sphere" below. |
42 The same goes for ``a $k$-sphere" below. |
43 |
43 |
44 Given a homeomorphism $f:X\to Y$ between $k$-balls, we want a corresponding |
44 |
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45 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
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46 the boundary), we want a corresponding |
45 bijection of sets $f:\cC(X)\to \cC(Y)$. |
47 bijection of sets $f:\cC(X)\to \cC(Y)$. |
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48 (This will imply ``strong duality", among other things.) |
46 So we replace the above with |
49 So we replace the above with |
47 |
50 |
48 \xxpar{Morphisms:} |
51 \xxpar{Morphisms:} |
49 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
52 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
50 the category of $k$-balls and |
53 the category of $k$-balls and |
53 (Note: We usually omit the subscript $k$.) |
56 (Note: We usually omit the subscript $k$.) |
54 |
57 |
55 We are being deliberately vague about what flavor of manifolds we are considering. |
58 We are being deliberately vague about what flavor of manifolds we are considering. |
56 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
59 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
57 They could be topological or PL or smooth. |
60 They could be topological or PL or smooth. |
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61 \nn{need to check whether this makes much difference --- see pseudo-isotopy below} |
58 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
62 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
59 to be fussier about corners.) |
63 to be fussier about corners.) |
60 For each flavor of manifold there is a corresponding flavor of $n$-category. |
64 For each flavor of manifold there is a corresponding flavor of $n$-category. |
61 We will concentrate of the case of PL unoriented manifolds. |
65 We will concentrate of the case of PL unoriented manifolds. |
62 |
66 |
63 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
67 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
64 of morphisms). |
68 of morphisms). |
65 The 0-sphere is unusual among spheres in that it is disconnected. |
69 The 0-sphere is unusual among spheres in that it is disconnected. |
66 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
70 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
67 (Actually, this is only true in the oriented case.) |
71 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
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72 by oriented 1-balls.) |
68 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
73 For $k>1$ and in the presence of strong duality the domain/range division makes less sense. |
69 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
74 \nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah} |
70 We prefer to combine the domain and range into a single entity which we call the |
75 We prefer to combine the domain and range into a single entity which we call the |
71 boundary of a morphism. |
76 boundary of a morphism. |
72 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
77 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
83 These maps, for various $X$, comprise a natural transformation of functors.} |
88 These maps, for various $X$, comprise a natural transformation of functors.} |
84 |
89 |
85 (Note that the first ``$\bd$" above is part of the data for the category, |
90 (Note that the first ``$\bd$" above is part of the data for the category, |
86 while the second is the ordinary boundary of manifolds.) |
91 while the second is the ordinary boundary of manifolds.) |
87 |
92 |
88 Given $c\in\cC(\bd(X))$, let $\cC(X; c) = \bd^{-1}(c)$. |
93 Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$. |
89 |
94 |
90 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
95 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
91 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
96 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
92 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
97 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
93 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
98 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
95 category structure. |
100 category structure. |
96 Note that this auxiliary structure is only in dimension $n$; |
101 Note that this auxiliary structure is only in dimension $n$; |
97 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
102 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
98 |
103 |
99 \medskip |
104 \medskip |
100 \nn{At the moment I'm a little confused about orientations, and more specifically |
105 \nn{ |
101 about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
106 %At the moment I'm a little confused about orientations, and more specifically |
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107 %about the role of orientation-reversing maps of boundaries when gluing oriented manifolds. |
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108 Maybe need a discussion about what the boundary of a manifold with a |
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109 structure (e.g. orientation) means. |
102 Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
110 Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold. |
103 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
111 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
104 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
112 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
105 equipped with an orientation of its once-stabilized tangent bundle. |
113 equipped with an orientation of its once-stabilized tangent bundle. |
106 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
114 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
123 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
131 \gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S) |
124 \] |
132 \] |
125 which is natural with respect to the actions of homeomorphisms.} |
133 which is natural with respect to the actions of homeomorphisms.} |
126 |
134 |
127 Note that we insist on injectivity above. |
135 Note that we insist on injectivity above. |
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136 |
128 Let $\cC(S)_E$ denote the image of $\gl_E$. |
137 Let $\cC(S)_E$ denote the image of $\gl_E$. |
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138 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
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139 |
129 We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as |
140 We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as |
130 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
141 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
131 |
142 |
132 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
143 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
133 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
144 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
156 (For $k=n$, see below.)} |
167 (For $k=n$, see below.)} |
157 |
168 |
158 \xxpar{Strict associativity:} |
169 \xxpar{Strict associativity:} |
159 {The composition (gluing) maps above are strictly associative.} |
170 {The composition (gluing) maps above are strictly associative.} |
160 |
171 |
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172 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
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173 |
161 The above two axioms are equivalent to the following axiom, |
174 The above two axioms are equivalent to the following axiom, |
162 which we state in slightly vague form. |
175 which we state in slightly vague form. |
163 |
176 |
164 \xxpar{Multi-composition:} |
177 \xxpar{Multi-composition:} |
165 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
178 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
177 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
190 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
178 \[ \xymatrix{ |
191 \[ \xymatrix{ |
179 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
192 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
180 X \ar[r]^{f} & X' |
193 X \ar[r]^{f} & X' |
181 } \] |
194 } \] |
182 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
195 commutes, then we have |
183 |
196 \[ |
184 \nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.} |
197 \tilde{f}(a\times D) = f(a)\times D' . |
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198 \] |
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199 Product morphisms are compatible with gluing (composition) in both factors: |
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200 \[ |
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201 (a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
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202 \] |
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203 and |
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204 \[ |
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205 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
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206 \] |
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207 Product morphisms are associative: |
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208 \[ |
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209 (a\times D)\times D' = a\times (D\times D') . |
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210 \] |
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211 (Here we are implicitly using functoriality and the obvious homeomorphism |
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212 $(X\times D)\times D' \to X\times(D\times D')$.) |
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213 } |
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214 |
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215 \nn{need even more subaxioms for product morphisms? |
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216 YES: need compatibility with certain restriction maps |
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217 in order to prove that dimension less than $n$ identities are act like identities.} |
185 |
218 |
186 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
219 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
187 The last axiom (below), concerning actions of |
220 The last axiom (below), concerning actions of |
188 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
221 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
189 |
222 |
252 a diagram like the one in Proposition \ref{CDprop} commutes. |
285 a diagram like the one in Proposition \ref{CDprop} commutes. |
253 \nn{repeat diagram here?} |
286 \nn{repeat diagram here?} |
254 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} |
287 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}} |
255 |
288 |
256 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
289 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
257 To do this we need to explain extended homeomorphisms form a space. |
290 To do this we need to explain how extended homeomorphisms form a topological space. |
258 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
291 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
259 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
292 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
260 \nn{need to also say something about collaring homeomorphisms.} |
293 \nn{need to also say something about collaring homeomorphisms.} |
261 \nn{this paragraph needs work.} |
294 \nn{this paragraph needs work.} |
262 |
295 |
279 Thus a system of fields determines an $n$-category simply by restricting our attention to |
312 Thus a system of fields determines an $n$-category simply by restricting our attention to |
280 balls. |
313 balls. |
281 The $n$-category can be thought of as the local part of the fields. |
314 The $n$-category can be thought of as the local part of the fields. |
282 Conversely, given an $n$-category we can construct a system of fields via |
315 Conversely, given an $n$-category we can construct a system of fields via |
283 \nn{gluing, or a universal construction} |
316 \nn{gluing, or a universal construction} |
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317 \nn{see subsection below} |
284 |
318 |
285 \nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
319 \nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
286 of fields. |
320 of fields. |
287 The universal (colimit) construction becomes our generalized definition of blob homology. |
321 The universal (colimit) construction becomes our generalized definition of blob homology. |
288 Need to explain how it relates to the old definition.} |
322 Need to explain how it relates to the old definition.} |
416 \subsection{Modules} |
450 \subsection{Modules} |
417 |
451 |
418 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
452 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
419 a.k.a.\ actions). |
453 a.k.a.\ actions). |
420 The definition will be very similar to that of $n$-categories. |
454 The definition will be very similar to that of $n$-categories. |
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455 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
421 |
456 |
422 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
457 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
423 in the context of an $m{+}1$-dimensional TQFT. |
458 in the context of an $m{+}1$-dimensional TQFT. |
424 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
459 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
425 This will be explained in more detail as we present the axioms. |
460 This will be explained in more detail as we present the axioms. |
626 |
661 |
627 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
662 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
628 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
663 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
629 component $\bd_i W$ of $W$. |
664 component $\bd_i W$ of $W$. |
630 |
665 |
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666 \nn{need to generalize to labeling codim 0 submanifolds of the boundary} |
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667 |
631 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
668 We will define a set $\cC(W, \cN)$ using a colimit construction similar to above. |
632 \nn{give ref} |
669 \nn{give ref} |
633 (If $k = n$ and our $k$-categories are enriched, then |
670 (If $k = n$ and our $k$-categories are enriched, then |
634 $\cC(W, \cN)$ will have additional structure; see below.) |
671 $\cC(W, \cN)$ will have additional structure; see below.) |
635 |
672 |
679 ** \nn{stuff below needs to be rewritten (shortened), because of new subsections above} |
716 ** \nn{stuff below needs to be rewritten (shortened), because of new subsections above} |
680 |
717 |
681 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
718 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
682 (If $k=1$ and manifolds are oriented, then one should be |
719 (If $k=1$ and manifolds are oriented, then one should be |
683 a left module and the other a right module.) |
720 a left module and the other a right module.) |
684 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depend (functorially) |
721 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially) |
685 on a choice of 1-ball (interval) $J$. |
722 on a choice of 1-ball (interval) $J$. |
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723 |
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724 |
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725 |
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726 |
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727 |
686 |
728 |
687 Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball |
729 Define a {\it doubly marked $k$-ball} to be a triple $(B, N, N')$, where $B$ is a $k$-ball |
688 and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$. |
730 and $N$ and $N'$ are disjoint $k{-}1$-balls in $\bd B$. |
689 |
731 |
690 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. |
732 Let $D = (B, N, N')$ be a doubly marked $k$-ball, $1\le k \le n$. |