397 \res_D\circ\pi^* = \rho^*\circ\res_Y . |
397 \res_D\circ\pi^* = \rho^*\circ\res_Y . |
398 \] |
398 \] |
399 \end{enumerate} |
399 \end{enumerate} |
400 } %%% end \noop %%% |
400 } %%% end \noop %%% |
401 \end{axiom} |
401 \end{axiom} |
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402 |
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403 To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
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404 Let $X$ be a $k$-ball and $Y\sub\bd X$ be a $(k{-}1)$-ball. |
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405 Let $J$ be a 1-ball. |
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406 Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
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407 A collar map is an instance of the composition |
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408 \[ |
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409 \cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
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410 \] |
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411 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
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412 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
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413 to the identity on the boundary. |
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414 |
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415 |
402 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
416 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
403 \label{axiom:extended-isotopies} |
417 \label{axiom:extended-isotopies} |
404 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
418 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
405 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
419 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
406 Then $f$ acts trivially on $\cC(X)$. |
420 Then $f$ acts trivially on $\cC(X)$. |
407 In addition, collar maps act trivially on $\cC(X)$. |
421 In addition, collar maps act trivially on $\cC(X)$. |
408 \end{axiom} |
422 \end{axiom} |
409 |
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410 \nn{need to define collar maps} |
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411 |
423 |
412 \smallskip |
424 \smallskip |
413 |
425 |
414 For $A_\infty$ $n$-categories, we replace |
426 For $A_\infty$ $n$-categories, we replace |
415 isotopy invariance with the requirement that families of homeomorphisms act. |
427 isotopy invariance with the requirement that families of homeomorphisms act. |