135 Note that we insist on injectivity above. |
135 Note that we insist on injectivity above. |
136 |
136 |
137 Let $\cC(S)_E$ denote the image of $\gl_E$. |
137 Let $\cC(S)_E$ denote the image of $\gl_E$. |
138 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
138 We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
139 |
139 |
140 We have ``restriction" maps $\cC(S)_E \to \cC(B_i)$, which can be thought of as |
140 We will call the projection $\cC(S)_E \to \cC(B_i)$ |
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141 a {\it restriction} map and write $\res_{B_i}(a)$ |
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142 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
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143 These restriction maps can be thought of as |
141 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
144 domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$. |
142 |
145 |
143 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
146 If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls |
144 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
147 as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$. |
145 |
148 |
168 |
171 |
169 \xxpar{Strict associativity:} |
172 \xxpar{Strict associativity:} |
170 {The composition (gluing) maps above are strictly associative.} |
173 {The composition (gluing) maps above are strictly associative.} |
171 |
174 |
172 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
175 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
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176 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
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177 a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
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178 Compositions of boundary and restriction maps will also be called restriction maps. |
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179 For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
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180 restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
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181 |
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182 %More notation and terminology: |
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183 %We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction} |
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184 %map |
173 |
185 |
174 The above two axioms are equivalent to the following axiom, |
186 The above two axioms are equivalent to the following axiom, |
175 which we state in slightly vague form. |
187 which we state in slightly vague form. |
176 |
188 |
177 \xxpar{Multi-composition:} |
189 \xxpar{Multi-composition:} |
208 \[ |
220 \[ |
209 (a\times D)\times D' = a\times (D\times D') . |
221 (a\times D)\times D' = a\times (D\times D') . |
210 \] |
222 \] |
211 (Here we are implicitly using functoriality and the obvious homeomorphism |
223 (Here we are implicitly using functoriality and the obvious homeomorphism |
212 $(X\times D)\times D' \to X\times(D\times D')$.) |
224 $(X\times D)\times D' \to X\times(D\times D')$.) |
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225 Product morphisms are compatible with restriction: |
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226 \[ |
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227 \res_{X\times E}(a\times D) = a\times E |
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228 \] |
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229 for $E\sub \bd D$ and $a\in \cC(X)$. |
213 } |
230 } |
214 |
231 |
215 \nn{need even more subaxioms for product morphisms? |
232 \nn{need even more subaxioms for product morphisms?} |
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.} |
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218 |
233 |
219 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
234 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
220 The last axiom (below), concerning actions of |
235 The last axiom (below), concerning actions of |
221 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
236 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
222 |
237 |
451 |
466 |
452 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
467 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
453 a.k.a.\ actions). |
468 a.k.a.\ actions). |
454 The definition will be very similar to that of $n$-categories. |
469 The definition will be very similar to that of $n$-categories. |
455 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
470 \nn{** need to make sure all revisions of $n$-cat def are also made to module def.} |
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471 \nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.} |
456 |
472 |
457 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
473 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
458 in the context of an $m{+}1$-dimensional TQFT. |
474 in the context of an $m{+}1$-dimensional TQFT. |
459 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
475 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
460 This will be explained in more detail as we present the axioms. |
476 This will be explained in more detail as we present the axioms. |
478 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
494 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
479 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
495 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
480 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
496 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
481 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
497 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
482 (The union is along $N\times \bd W$.) |
498 (The union is along $N\times \bd W$.) |
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499 (If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
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500 the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
483 |
501 |
484 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
502 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
485 Call such a thing a {marked $k{-}1$-hemisphere}. |
503 Call such a thing a {marked $k{-}1$-hemisphere}. |
486 |
504 |
487 \xxpar{Module boundaries, part 1:} |
505 \xxpar{Module boundaries, part 1:} |
493 |
511 |
494 \xxpar{Module boundaries, part 2:} |
512 \xxpar{Module boundaries, part 2:} |
495 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
513 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
496 These maps, for various $M$, comprise a natural transformation of functors.} |
514 These maps, for various $M$, comprise a natural transformation of functors.} |
497 |
515 |
498 Given $c\in\cM(\bd M)$, let $\cM(M; c) = \bd^{-1}(c)$. |
516 Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
499 |
517 |
500 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
518 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
501 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
519 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
502 and $c\in \cC(\bd M)$. |
520 and $c\in \cC(\bd M)$. |
503 |
521 |
510 \[ |
528 \[ |
511 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
529 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
512 \] |
530 \] |
513 which is natural with respect to the actions of homeomorphisms.} |
531 which is natural with respect to the actions of homeomorphisms.} |
514 |
532 |
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533 Let $\cM(H)_E$ denote the image of $\gl_E$. |
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534 We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
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535 |
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536 |
515 \xxpar{Axiom yet to be named:} |
537 \xxpar{Axiom yet to be named:} |
516 {For each marked $k$-hemisphere $H$ there is a restriction map |
538 {For each marked $k$-hemisphere $H$ there is a restriction map |
517 $\cM(H)\to \cC(H)$. |
539 $\cM(H)\to \cC(H)$. |
518 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
540 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
519 These maps comprise a natural transformation of functors.} |
541 These maps comprise a natural transformation of functors.} |
520 |
542 |
521 Note that combining the various boundary and restriction maps above |
543 Note that combining the various boundary and restriction maps above |
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544 (for both modules and $n$-categories) |
522 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
545 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
523 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
546 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
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547 The subset is the subset of morphisms which are appropriately splittable (transverse to the |
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548 cutting submanifolds). |
524 This fact will be used below. |
549 This fact will be used below. |
525 \nn{need to say more about splitableness/transversality in various places above} |
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526 |
550 |
527 In our example, the various restriction and gluing maps above come from |
551 In our example, the various restriction and gluing maps above come from |
528 restricting and gluing maps into $T$. |
552 restricting and gluing maps into $T$. |
529 |
553 |
530 We require two sorts of composition (gluing) for modules, corresponding to two ways |
554 We require two sorts of composition (gluing) for modules, corresponding to two ways |
570 (For $k=n$, see below.)} |
594 (For $k=n$, see below.)} |
571 |
595 |
572 \xxpar{Module strict associativity:} |
596 \xxpar{Module strict associativity:} |
573 {The composition and action maps above are strictly associative.} |
597 {The composition and action maps above are strictly associative.} |
574 |
598 |
575 The above two axioms are equivalent to the following axiom, |
599 Note that the above associativity axiom applies to mixtures of module composition, |
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600 action maps and $n$-category composition. |
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601 See Figure xxxx. |
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602 |
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603 The above three axioms are equivalent to the following axiom, |
576 which we state in slightly vague form. |
604 which we state in slightly vague form. |
577 \nn{need figure for this} |
605 \nn{need figure for this} |
578 |
606 |
579 \xxpar{Module multi-composition:} |
607 \xxpar{Module multi-composition:} |
580 {Given any decomposition |
608 {Given any decomposition |
605 M \ar[r]^{f} & M' |
633 M \ar[r]^{f} & M' |
606 } \] |
634 } \] |
607 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
635 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
608 |
636 |
609 \nn{Need to say something about compatibility with gluing (of both $M$ and $D$) above.} |
637 \nn{Need to say something about compatibility with gluing (of both $M$ and $D$) above.} |
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638 |
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639 \nn{** marker --- resume revising here **} |
610 |
640 |
611 There are two alternatives for the next axiom, according whether we are defining |
641 There are two alternatives for the next axiom, according whether we are defining |
612 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
642 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
613 In the plain case we require |
643 In the plain case we require |
614 |
644 |