549 where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
549 where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch |
550 above the non-marked boundary component of $J$. |
550 above the non-marked boundary component of $J$. |
551 \nn{give figure for this, or say more?} |
551 \nn{give figure for this, or say more?} |
552 Then $\cE$ has the structure of an $n{-}1$-category. |
552 Then $\cE$ has the structure of an $n{-}1$-category. |
553 |
553 |
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554 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
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555 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
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556 In this case ($k=1$ and oriented or Spin), there are two types |
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557 of marked 1-balls, call them left-marked and right-marked, |
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558 and hence there are two types of modules, call them right modules and left modules. |
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559 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
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560 there is no left/right module distinction. |
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561 |
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562 \medskip |
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563 |
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564 Next we consider tensor products (or, more generally, self tensor products |
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565 or coends). |
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566 |
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567 |
554 |
568 |
555 \medskip |
569 \medskip |
556 \hrule |
570 \hrule |
557 \medskip |
571 \medskip |
558 |
572 |
562 |
576 |
563 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
577 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
564 a separate paper): |
578 a separate paper): |
565 \begin{itemize} |
579 \begin{itemize} |
566 \item tensor products |
580 \item tensor products |
567 \item blob complex is an example of an $A_\infty$ $n$-category |
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568 \item fundamental $n$-groupoid is an example of an $A_\infty$ $n$-category |
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569 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
581 \item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat |
570 \item conversely, our def implies other defs |
582 \item conversely, our def implies other defs |
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583 \item do same for modules; maybe an appendix on relating topological |
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584 vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
571 \item traditional $A_\infty$ 1-cat def implies our def |
585 \item traditional $A_\infty$ 1-cat def implies our def |
572 \item ... and vice-versa (already done in appendix) |
586 \item ... and vice-versa (already done in appendix) |
573 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
587 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
574 \item spell out what difference (if any) Top vs PL vs Smooth makes |
588 \item spell out what difference (if any) Top vs PL vs Smooth makes |
575 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
589 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
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590 \item define $n{+}1$-cat of $n$-cats; discuss Morita equivalence |
576 \end{itemize} |
591 \end{itemize} |
577 |
592 |
578 |
593 |