290 $$\mathfig{.8}{tempkw/blah7}$$ |
290 $$\mathfig{.8}{tempkw/blah7}$$ |
291 \caption{Operad composition and associativity}\label{blah7}\end{figure} |
291 \caption{Operad composition and associativity}\label{blah7}\end{figure} |
292 |
292 |
293 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
293 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
294 |
294 |
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295 \begin{axiom}[Product (identity) morphisms, preliminary version] |
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296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
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297 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
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298 These maps must satisfy the following conditions. |
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299 \begin{enumerate} |
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300 \item |
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301 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
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302 \[ \xymatrix{ |
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303 X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
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304 X \ar[r]^{f} & X' |
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305 } \] |
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306 commutes, then we have |
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307 \[ |
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308 \tilde{f}(a\times D) = f(a)\times D' . |
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309 \] |
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310 \item |
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311 Product morphisms are compatible with gluing (composition) in both factors: |
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312 \[ |
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313 (a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
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314 \] |
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315 and |
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316 \[ |
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317 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
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318 \] |
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319 \item |
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320 Product morphisms are associative: |
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321 \[ |
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322 (a\times D)\times D' = a\times (D\times D') . |
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323 \] |
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324 (Here we are implicitly using functoriality and the obvious homeomorphism |
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325 $(X\times D)\times D' \to X\times(D\times D')$.) |
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326 \item |
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327 Product morphisms are compatible with restriction: |
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328 \[ |
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329 \res_{X\times E}(a\times D) = a\times E |
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330 \] |
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331 for $E\sub \bd D$ and $a\in \cC(X)$. |
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332 \end{enumerate} |
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333 \end{axiom} |
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334 |
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335 We will need to strengthen the above preliminary version of the axiom to allow |
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336 for products which are ``pinched" in various ways along their boundary. |
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337 (See Figure xxxx.) |
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338 (The need for a strengthened version will become apparent in appendix \ref{sec:comparing-defs}.) |
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339 Define a {\it pinched product} to be a map |
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340 \[ |
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341 \pi: E\to X |
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342 \] |
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343 such that $E$ is an $m$-ball, $X$ is a $k$-ball ($k<m$), and $\pi$ is locally modeled |
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344 on a standard iterated degeneracy map |
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345 \[ |
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346 d: \Delta^m\to\Delta^k . |
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347 \] |
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348 In other words, \nn{each point has a neighborhood blah blah...} |
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349 (We thank Kevin Costello for suggesting this approach.) |
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350 |
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351 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m{-}k$-ball, |
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352 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
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353 $l \le m-k$. |
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354 |
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355 It is easy to see that a composition of pinched products is again a pinched product. |
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356 |
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357 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
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358 $\pi:E'\to \pi(E')$ is again a pinched product. |
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359 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
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360 such that each $E_i\sub E$ is a sub pinched product. |
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361 (See Figure xxxx.) |
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362 |
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363 The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
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364 $\pi:E\to X$. |
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365 Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
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366 In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
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367 In the case where $\cC(X)$ is the set of all labeled embedded cell complexes in $X$, |
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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. |
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369 |
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370 \nn{resume revising here; need to rewrite the following pasted axiom} |
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371 |
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372 \addtocounter{axiom}{-1} |
295 \begin{axiom}[Product (identity) morphisms] |
373 \begin{axiom}[Product (identity) morphisms] |
296 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
374 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
297 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
375 usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
298 These maps must satisfy the following conditions. |
376 These maps must satisfy the following conditions. |
299 \begin{enumerate} |
377 \begin{enumerate} |
330 \] |
408 \] |
331 for $E\sub \bd D$ and $a\in \cC(X)$. |
409 for $E\sub \bd D$ and $a\in \cC(X)$. |
332 \end{enumerate} |
410 \end{enumerate} |
333 \end{axiom} |
411 \end{axiom} |
334 |
412 |
335 \nn{need even more subaxioms for product morphisms?} |
413 |
336 |
414 |
337 \nn{Almost certainly we need a little more than the above axiom. |
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338 More specifically, in order to bootstrap our way from the top dimension |
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339 properties of identity morphisms to low dimensions, we need regular products, |
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340 pinched products and even half-pinched products. |
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341 I'm not sure what the best way to cleanly axiomatize the properties of these various |
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342 products is. |
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343 For the moment, I'll assume that all flavors of the product are at |
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344 our disposal, and I'll plan on revising the axioms later.} |
422 \medskip |
345 |
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346 \nn{current idea for fixing this: make the above axiom a ``preliminary version" |
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347 (as we have already done with some of the other axioms), then state the official |
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348 axiom for maps $\pi: E \to X$ which are almost fiber bundles. |
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349 one option is to restrict E to be a (full/half/not)-pinched product (up to homeo). |
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350 the alternative is to give some sort of local criterion for what's allowed. |
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351 state a gluing axiom for decomps $E = E'\cup E''$ where all three are of the correct type. |
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352 } |
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353 |
423 |
354 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
424 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
355 The last axiom (below), concerning actions of |
425 The last axiom (below), concerning actions of |
356 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
426 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
357 |
427 |