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236 |
236 |
237 \begin{axiom}[Boundaries]\label{nca-boundary} |
237 \begin{axiom}[Boundaries]\label{nca-boundary} |
238 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
238 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
239 These maps, for various $X$, comprise a natural transformation of functors. |
239 These maps, for various $X$, comprise a natural transformation of functors. |
240 \end{axiom} |
240 \end{axiom} |
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241 |
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242 For $c\in \cl{\cC}_{k-1}(\bd X)$ we let $\cC_k(X; c)$ denote the preimage $\bd^{-1}(c)$. |
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243 |
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244 Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
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245 (e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces). |
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246 This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
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247 of an object of $\cS$, and all of the structure maps of the category (above and below) are |
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248 compatible with the $\cS$ structure on $\cC_n(X; c)$. |
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249 |
241 |
250 |
242 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
251 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere. |
243 |
252 |
244 \begin{lem} |
253 \begin{lem} |
245 \label{lem:domain-and-range} |
254 \label{lem:domain-and-range} |
371 |
380 |
372 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
381 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
373 Maybe just a single remark that we are omitting some details which appear in our |
382 Maybe just a single remark that we are omitting some details which appear in our |
374 longer paper.} |
383 longer paper.} |
375 \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.} |
384 \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.} |
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385 \nn{KW: It's not the length I'm worried about --- I was worried about distracting the reader |
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386 with an arcane technical issue. But we can decide later.} |
376 |
387 |
377 A \emph{ball decomposition} of $W$ is a |
388 A \emph{ball decomposition} of $W$ is a |
378 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
389 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
379 $\du_a X_a$ and each $M_i$ is a manifold. |
390 $\du_a X_a$ and each $M_i$ is a manifold. |
380 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
391 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |