25 model the $k$-morphisms on more complicated combinatorial polyhedra. |
25 model the $k$-morphisms on more complicated combinatorial polyhedra. |
26 |
26 |
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):}{For any $k$-manifold $X$ homeomorphic |
30 \xxpar{Morphisms (preliminary version):} |
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31 {For any $k$-manifold $X$ homeomorphic |
31 to a $k$-ball, we have a set of $k$-morphisms |
32 to a $k$-ball, we have a set of $k$-morphisms |
32 $\cC(X)$.} |
33 $\cC(X)$.} |
33 |
34 |
34 Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding |
35 Given a homeomorphism $f:X\to Y$ between such $k$-manifolds, we want a corresponding |
35 bijection of sets $f:\cC(X)\to \cC(Y)$. |
36 bijection of sets $f:\cC(X)\to \cC(Y)$. |
36 So we replace the above with |
37 So we replace the above with |
37 |
38 |
38 \xxpar{Morphisms:}{For each $0 \le k \le n$, we have a functor $\cC_k$ from |
39 \xxpar{Morphisms:} |
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40 {For each $0 \le k \le n$, we have a functor $\cC_k$ from |
39 the category of manifolds homeomorphic to the $k$-ball and |
41 the category of manifolds homeomorphic to the $k$-ball and |
40 homeomorphisms to the category of sets and bijections.} |
42 homeomorphisms to the category of sets and bijections.} |
41 |
43 |
42 (Note: We usually omit the subscript $k$.) |
44 (Note: We usually omit the subscript $k$.) |
43 |
45 |
142 to the intersection of the boundaries of $B$ and $B_i$. |
144 to the intersection of the boundaries of $B$ and $B_i$. |
143 If $k < n$ we require that $\gl_Y$ is injective. |
145 If $k < n$ we require that $\gl_Y$ is injective. |
144 (For $k=n$, see below.)} |
146 (For $k=n$, see below.)} |
145 |
147 |
146 \xxpar{Strict associativity:} |
148 \xxpar{Strict associativity:} |
147 {The composition (gluing) maps above are strictly associative. |
149 {The composition (gluing) maps above are strictly associative.} |
148 It follows that given a decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
150 |
149 into small $k$-balls, there is a well-defined |
151 The above two axioms are equivalent to the following axiom, |
150 map from an appropriate subset of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
152 which we state in slightly vague form. |
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153 |
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154 \xxpar{Multi-composition:} |
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155 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
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156 into small $k$-balls, there is a |
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157 map from an appropriate subset (like a fibered product) |
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158 of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
151 and these various $m$-fold composition maps satisfy an |
159 and these various $m$-fold composition maps satisfy an |
152 operad-type associativity condition.} |
160 operad-type strict associativity condition.} |
153 |
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154 \nn{above maybe needs some work} |
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155 |
161 |
156 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
162 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
157 |
163 |
158 \xxpar{Product (identity) morphisms:} |
164 \xxpar{Product (identity) morphisms:} |
159 {Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$. |
165 {Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$. |
304 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
310 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
305 combinations of $C$-labeled sub cell complexes of $X$ |
311 combinations of $C$-labeled sub cell complexes of $X$ |
306 modulo the kernel of the evaluation map. |
312 modulo the kernel of the evaluation map. |
307 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
313 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
308 and with the same labeling as $a$. |
314 and with the same labeling as $a$. |
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315 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
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316 Define $\cC(X)$, for $\dim(X) < n$, |
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317 to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
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318 Define $\cC(X; c)$, for $X$ an $n$-ball, |
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319 to be the dual Hilbert space $A(X\times F; c)$. |
309 \nn{refer elsewhere for details?} |
320 \nn{refer elsewhere for details?} |
310 |
321 |
311 \item Variation on the above examples: |
322 \item Variation on the above examples: |
312 We could allow $F$ to have boundary and specify boundary conditions on $(\bd X)\times F$, |
323 We could allow $F$ to have boundary and specify boundary conditions on $(\bd X)\times F$, |
313 for example product boundary conditions or take the union over all boundary conditions. |
324 for example product boundary conditions or take the union over all boundary conditions. |
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325 \nn{maybe should not emphasize this case, since it's ``better" in some sense |
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326 to think of these guys as affording a representation |
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327 of the $n{+}1$-category associated to $\bd F$.} |
314 |
328 |
315 \end{itemize} |
329 \end{itemize} |
316 |
330 |
317 |
331 |
318 Examples of $A_\infty$ $n$-categories: |
332 Examples of $A_\infty$ $n$-categories: |
332 |
346 |
333 \medskip |
347 \medskip |
334 |
348 |
335 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
349 Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations, |
336 a.k.a.\ actions). |
350 a.k.a.\ actions). |
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351 The definition will be very similar to that of $n$-categories. |
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352 |
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353 Out motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
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354 in the context of an $m{+}1$-dimensional TQFT. |
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355 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
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356 This will be explained in more detail as we present the axioms. |
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357 |
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358 Fix an $n$-category $\cC$. |
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359 |
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360 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
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361 (standard $k$-ball, northern hemisphere in boundary of standard $k$-ball). |
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362 We call $B$ the ball and $N$ the marking. |
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363 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
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364 restricts to a homeomorphism of markings. |
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365 |
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366 \xxpar{Module morphisms} |
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367 {For each $0 \le k \le n$, we have a functor $\cM_k$ from |
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368 the category of marked $k$-balls and |
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369 homeomorphisms to the category of sets and bijections.} |
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370 |
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371 (As with $n$-categories, we will usually omit the subscript $k$.) |
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372 |
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373 In our example, let $\cM(B, N) = \cD((B\times \bd W)\cup_{N\times \bd W} (N\times W))$, |
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374 where $\cD$ is the fields functor for the TQFT. |
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375 |
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376 Define the boundary of a marked ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
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377 Call such a thing a {marked hemisphere}. |
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378 |
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379 \xxpar{Module boundaries, part 1:} |
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380 {For each $0 \le k \le n-1$, we have a functor $\cM_k$ from |
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381 the category of marked hemispheres (of dimension $k$) and |
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382 homeomorphisms to the category of sets and bijections.} |
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383 |
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384 \xxpar{Module boundaries, part 2:} |
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385 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$. |
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386 These maps, for various $M$, comprise a natural transformation of functors.} |
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387 |
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388 Given $c\in\cM(\bd M)$, let $\cM(M; c) = \bd^{-1}(c)$. |
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389 |
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390 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
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391 then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
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392 and $c\in \cC(\bd M)$. |
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393 |
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394 \xxpar{Module domain $+$ range $\to$ boundary:} |
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395 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$), |
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396 $B_i$ is a marked $k$-ball, and $E = B_1\cap B_2$ is a marked $k{-}1$-hemisphere. |
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397 Let $\cM(B_1) \times_{\cM(E)} \cM(B_2)$ denote the fibered product of the |
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398 two maps $\bd: \cM(B_i)\to \cM(E)$. |
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399 Then (axiom) we have an injective map |
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400 \[ |
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401 \gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H) |
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402 \] |
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403 which is natural with respect to the actions of homeomorphisms.} |
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404 |
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405 |
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406 |
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407 |
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408 |
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409 |
337 |
410 |
338 \medskip |
411 \medskip |
339 \hrule |
412 \hrule |
340 \medskip |
413 \medskip |
341 |
414 |
344 |
417 |
345 |
418 |
346 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
419 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
347 a separate paper): |
420 a separate paper): |
348 \begin{itemize} |
421 \begin{itemize} |
349 \item modules/representations/actions (need to decide what to call them) |
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350 \item tensor products |
422 \item tensor products |
351 \item blob complex is an example of an $A_\infty$ $n$-category |
423 \item blob complex is an example of an $A_\infty$ $n$-category |
352 \item fundamental $n$-groupoid is an example of an $A_\infty$ $n$-category |
424 \item fundamental $n$-groupoid is an example of an $A_\infty$ $n$-category |
353 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
425 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
354 \item conversely, our def implies other defs |
426 \item conversely, our def implies other defs |