212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
213 a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
213 a product of $k$ intervals \nn{cf xxxx} but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
214 to the standard $k$-ball $B^k$. |
214 to the standard $k$-ball $B^k$. |
215 \nn{maybe add that in addition we want functoriality} |
215 \nn{maybe add that in addition we want functoriality} |
216 |
216 |
217 In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs. |
217 We haven't said precisely what sort of balls we are considering, |
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218 because we prefer to let this detail be a parameter in the definition. |
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219 It is useful to consider unoriented, oriented, Spin and $\mbox{Pin}_\pm$ balls. |
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220 Also useful are more exotic structures, such as balls equipped with a map to some target space, |
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221 or equipped with $m$ independent vector fields. |
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222 (The latter structure would model $n$-categories with less duality than we usually assume.) |
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223 |
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224 %In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs. |
218 |
225 |
219 \begin{axiom}[Morphisms] |
226 \begin{axiom}[Morphisms] |
220 \label{axiom:morphisms} |
227 \label{axiom:morphisms} |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
228 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
222 the category of $k$-balls and |
229 the category of $k$-balls and |
353 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
360 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
354 Then $f$ acts trivially on $\cC(X)$. |
361 Then $f$ acts trivially on $\cC(X)$. |
355 In addition, collar maps act trivially on $\cC(X)$. |
362 In addition, collar maps act trivially on $\cC(X)$. |
356 \end{axiom} |
363 \end{axiom} |
357 |
364 |
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365 \nn{need to define collar maps} |
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366 |
358 \smallskip |
367 \smallskip |
359 |
368 |
360 For $A_\infty$ $n$-categories, we replace |
369 For $A_\infty$ $n$-categories, we replace |
361 isotopy invariance with the requirement that families of homeomorphisms act. |
370 isotopy invariance with the requirement that families of homeomorphisms act. |
362 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
371 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
379 \todo{ |
388 \todo{ |
380 Decide if we need a friendlier, skein-module version. |
389 Decide if we need a friendlier, skein-module version. |
381 } |
390 } |
382 |
391 |
383 \subsubsection{Examples} |
392 \subsubsection{Examples} |
384 \todo{maps to a space, string diagrams} |
393 |
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394 \nn{can't figure out environment stuff; want no italics} |
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395 |
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396 \noindent |
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397 Example. [Fundamental $n$-groupoid of a space] |
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398 Let $T$ be a topological space. |
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399 Define $\pi_{\le n}(T)(X)$, for $X$ a $k$-ball and $k<n$, |
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400 to be the set of continuous maps from $X$ to $T$. |
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401 If $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
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402 Define boundary restrictions and gluing in the obvious way. |
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403 If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism, |
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404 define the product morphism $\rho^*(f)$ to be $f\circ\rho$. |
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405 |
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406 We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
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407 Most of the definition is the same as above. |
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408 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
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409 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
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410 |
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411 |
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412 \noindent |
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413 Example. [String diagrams] |
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414 Fix a traditional $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
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415 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
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416 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
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417 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
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418 Boundary restrictions and gluing are again straightforward to define. |
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419 Define product morphisms via product cell decompositions. |
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420 |
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421 |
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422 \nn{also do bordism category?} |
385 |
423 |
386 \subsection{The blob complex} |
424 \subsection{The blob complex} |
387 \subsubsection{Decompositions of manifolds} |
425 \subsubsection{Decompositions of manifolds} |
388 |
426 |
389 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
427 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
390 Maybe just a single remark that we are omitting some details which appear in our |
428 Maybe just a single remark that we are omitting some details which appear in our |
391 longer paper.} |
429 longer paper.} |
392 \nn{SM: for now I disagree: the space expense is pretty minor, and it always us to be "in principle" complete. Let's see how we go for length.} |
430 \nn{SM: for now I disagree: the space expense is pretty minor, and it allows us to be ``in principle" complete. Let's see how we go for length.} |
393 \nn{KW: It's not the length I'm worried about --- I was worried about distracting the reader |
431 \nn{KW: It's not the length I'm worried about --- I was worried about distracting the reader |
394 with an arcane technical issue. But we can decide later.} |
432 with an arcane technical issue. But we can decide later.} |
395 |
433 |
396 A \emph{ball decomposition} of $W$ is a |
434 A \emph{ball decomposition} of $W$ is a |
397 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
435 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |