355 |
355 |
356 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
356 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
357 a.k.a.\ actions). |
357 a.k.a.\ actions). |
358 The definition will be very similar to that of $n$-categories. |
358 The definition will be very similar to that of $n$-categories. |
359 |
359 |
360 Out motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
360 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
361 in the context of an $m{+}1$-dimensional TQFT. |
361 in the context of an $m{+}1$-dimensional TQFT. |
362 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
362 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
363 This will be explained in more detail as we present the axioms. |
363 This will be explained in more detail as we present the axioms. |
364 |
364 |
365 Fix an $n$-category $\cC$. |
365 Fix an $n$-category $\cC$. |
375 the category of marked $k$-balls and |
375 the category of marked $k$-balls and |
376 homeomorphisms to the category of sets and bijections.} |
376 homeomorphisms to the category of sets and bijections.} |
377 |
377 |
378 (As with $n$-categories, we will usually omit the subscript $k$.) |
378 (As with $n$-categories, we will usually omit the subscript $k$.) |
379 |
379 |
380 In our example, let $\cM(B, N) = \cD((B\times \bd W)\cup_{N\times \bd W} (N\times W))$, |
380 For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set |
381 where $\cD$ is the fields functor for the TQFT. |
381 of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$. |
|
382 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
|
383 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
384 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
|
385 (The union is along $N\times \bd W$.) |
382 |
386 |
383 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
387 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
384 Call such a thing a {marked $k{-}1$-hemisphere}. |
388 Call such a thing a {marked $k{-}1$-hemisphere}. |
385 |
389 |
386 \xxpar{Module boundaries, part 1:} |
390 \xxpar{Module boundaries, part 1:} |
387 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
391 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
388 the category of marked hemispheres (of dimension $k$) and |
392 the category of marked $k$-hemispheres and |
389 homeomorphisms to the category of sets and bijections.} |
393 homeomorphisms to the category of sets and bijections.} |
|
394 |
|
395 In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. |
390 |
396 |
391 \xxpar{Module boundaries, part 2:} |
397 \xxpar{Module boundaries, part 2:} |
392 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
398 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
393 These maps, for various $M$, comprise a natural transformation of functors.} |
399 These maps, for various $M$, comprise a natural transformation of functors.} |
394 |
400 |
398 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
404 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
399 and $c\in \cC(\bd M)$. |
405 and $c\in \cC(\bd M)$. |
400 |
406 |
401 \xxpar{Module domain $+$ range $\to$ boundary:} |
407 \xxpar{Module domain $+$ range $\to$ boundary:} |
402 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
408 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
403 $B_i$ is a marked $k$-ball, and $E = B_1\cap B_2$ is a marked $k{-}1$-hemisphere. |
409 $M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere. |
404 Let $\cM(B_1) \times_{\cM(E)} \cM(B_2)$ denote the fibered product of the |
410 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
405 two maps $\bd: \cM(B_i)\to \cM(E)$. |
411 two maps $\bd: \cM(M_i)\to \cM(E)$. |
406 Then (axiom) we have an injective map |
412 Then (axiom) we have an injective map |
407 \[ |
413 \[ |
408 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
414 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
409 \] |
415 \] |
410 which is natural with respect to the actions of homeomorphisms.} |
416 which is natural with respect to the actions of homeomorphisms.} |
419 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
425 we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
420 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
426 a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
421 This fact will be used below. |
427 This fact will be used below. |
422 \nn{need to say more about splitableness/transversality in various places above} |
428 \nn{need to say more about splitableness/transversality in various places above} |
423 |
429 |
424 We stipulate two sorts of composition (gluing) for modules, corresponding to two ways |
430 In our example, the various restriction and gluing maps above come from |
|
431 restricting and gluing maps into $T$. |
|
432 |
|
433 We require two sorts of composition (gluing) for modules, corresponding to two ways |
425 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
434 of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
426 First, we can compose two module morphisms to get another module morphism. |
435 First, we can compose two module morphisms to get another module morphism. |
427 |
436 |
428 \nn{need figures for next two axioms} |
437 \nn{need figures for next two axioms} |
429 |
438 |
532 \nn{repeat diagram here?} |
541 \nn{repeat diagram here?} |
533 \nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
542 \nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}} |
534 |
543 |
535 \medskip |
544 \medskip |
536 |
545 |
537 |
546 Note that the above axioms imply that an $n$-category module has the structure |
|
547 of an $n{-}1$-category. |
|
548 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
549 where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
|
550 above the non-marked boundary component of $J$. |
|
551 \nn{give figure for this, or say more?} |
|
552 Then $\cE$ has the structure of an $n{-}1$-category. |
538 |
553 |
539 |
554 |
540 \medskip |
555 \medskip |
541 \hrule |
556 \hrule |
542 \medskip |
557 \medskip |