333 \end{axiom} |
333 \end{axiom} |
334 |
334 |
335 We will need to strengthen the above preliminary version of the axiom to allow |
335 We will need to strengthen the above preliminary version of the axiom to allow |
336 for products which are ``pinched" in various ways along their boundary. |
336 for products which are ``pinched" in various ways along their boundary. |
337 (See Figure xxxx.) |
337 (See Figure xxxx.) |
338 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs}.) |
338 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs} |
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339 where we construct a traditional category from a topological category.) |
339 Define a {\it pinched product} to be a map |
340 Define a {\it pinched product} to be a map |
340 \[ |
341 \[ |
341 \pi: E\to X |
342 \pi: E\to X |
342 \] |
343 \] |
343 such that $E$ is an $m$-ball, $X$ is a $k$-ball ($k<m$), and $\pi$ is locally modeled |
344 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
344 on a standard iterated degeneracy map |
345 on a standard iterated degeneracy map |
345 \[ |
346 \[ |
346 d: \Delta^m\to\Delta^k . |
347 d: \Delta^{k+m}\to\Delta^k . |
347 \] |
348 \] |
348 In other words, \nn{each point has a neighborhood blah blah...} |
349 In other words, \nn{each point has a neighborhood blah blah...} |
349 (We thank Kevin Costello for suggesting this approach.) |
350 (We thank Kevin Costello for suggesting this approach.) |
350 |
351 |
351 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m{-}k$-ball, |
352 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
352 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
353 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
353 $l \le m-k$. |
354 $l \le m$, with $l$ depending on $x$. |
354 |
355 |
355 It is easy to see that a composition of pinched products is again a pinched product. |
356 It is easy to see that a composition of pinched products is again a pinched product. |
356 |
357 |
357 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
358 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
358 $\pi:E'\to \pi(E')$ is again a pinched product. |
359 $\pi:E'\to \pi(E')$ is again a pinched product. |
360 such that each $E_i\sub E$ is a sub pinched product. |
361 such that each $E_i\sub E$ is a sub pinched product. |
361 (See Figure xxxx.) |
362 (See Figure xxxx.) |
362 |
363 |
363 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
364 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
364 $\pi:E\to X$. |
365 $\pi:E\to X$. |
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366 Morphisms in the image of $\pi^*$ will be called product morphisms. |
365 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
367 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
366 In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
368 In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
367 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes in $X$, |
369 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
368 taking inverse images under $\pi$ gives a cell decomposition of $E$ in which corresponding cells have the same codimension, and therefore accept the same labels. |
370 define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
369 |
371 same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
370 \nn{resume revising here; need to rewrite the following pasted axiom} |
372 |
371 |
373 |
372 \addtocounter{axiom}{-1} |
374 \addtocounter{axiom}{-1} |
373 \begin{axiom}[Product (identity) morphisms] |
375 \begin{axiom}[Product (identity) morphisms] |
374 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
376 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
375 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
377 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
376 These maps must satisfy the following conditions. |
378 These maps must satisfy the following conditions. |
377 \begin{enumerate} |
379 \begin{enumerate} |
378 \item |
380 \item |
379 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
381 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
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382 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
380 \[ \xymatrix{ |
383 \[ \xymatrix{ |
381 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
384 E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
382 X \ar[r]^{f} & X' |
385 X \ar[r]^{f} & X' |
383 } \] |
386 } \] |
384 commutes, then we have |
387 commutes, then we have |
385 \[ |
388 \[ |
386 \tilde{f}(a\times D) = f(a)\times D' . |
389 \pi'^*\circ f = \tilde{f}\circ \pi^*. |
387 \] |
390 \] |
388 \item |
391 \item |
389 Product morphisms are compatible with gluing (composition) in both factors: |
392 Product morphisms are compatible with gluing (composition). |
390 \[ |
393 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
391 (a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
394 be pinched products with $E = E_1\cup E_2$. |
392 \] |
395 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
393 and |
396 Then |
394 \[ |
397 \[ |
395 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
398 \pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
396 \] |
399 \] |
397 \item |
400 \item |
398 Product morphisms are associative: |
401 Product morphisms are associative. |
399 \[ |
402 If $\pi:E\to X$ and $\rho:D\to E$ and pinched products then |
400 (a\times D)\times D' = a\times (D\times D') . |
403 \[ |
401 \] |
404 \rho^*\circ\pi^* = (\pi\circ\rho)^* . |
402 (Here we are implicitly using functoriality and the obvious homeomorphism |
405 \] |
403 $(X\times D)\times D' \to X\times(D\times D')$.) |
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404 \item |
406 \item |
405 Product morphisms are compatible with restriction: |
407 Product morphisms are compatible with restriction. |
406 \[ |
408 If we have a commutative diagram |
407 \res_{X\times E}(a\times D) = a\times E |
409 \[ \xymatrix{ |
408 \] |
410 D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
409 for $E\sub \bd D$ and $a\in \cC(X)$. |
411 Y \ar@{^(->}[r] & X |
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412 } \] |
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413 such that $\rho$ and $\pi$ are pinched products, then |
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414 \[ |
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415 \res_D\circ\pi^* = \rho^*\circ\res_Y . |
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416 \] |
410 \end{enumerate} |
417 \end{enumerate} |
411 \end{axiom} |
418 \end{axiom} |
412 |
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413 |
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414 |
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415 |
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416 |
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417 |
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418 |
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419 |
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420 |
419 |
421 |
420 |
422 \medskip |
421 \medskip |
423 |
422 |
424 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
423 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |