equal
deleted
inserted
replaced
455 In the plain (non-$A_\infty$) case, this means that |
455 In the plain (non-$A_\infty$) case, this means that |
456 for each decomposition $x$ there is a map |
456 for each decomposition $x$ there is a map |
457 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
457 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
458 above, and $\cC(W)$ is universal with respect to these properties. |
458 above, and $\cC(W)$ is universal with respect to these properties. |
459 In the $A_\infty$ case, it means |
459 In the $A_\infty$ case, it means |
460 \nn{.... need to check if there is a def in the literature before writing this down} |
460 \nn{.... need to check if there is a def in the literature before writing this down; |
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461 homotopy colimit I think} |
461 |
462 |
462 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
463 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
463 \[ |
464 \[ |
464 \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
465 \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
465 \] |
466 \] |
467 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
468 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
468 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
469 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
469 |
470 |
470 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
471 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
471 is as follows. |
472 is as follows. |
|
473 \nn{should probably rewrite this to be compatible with some standard reference} |
472 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
474 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
473 Such sequences (for all $m$) form a simplicial set. |
475 Such sequences (for all $m$) form a simplicial set. |
474 Let |
476 Let |
475 \[ |
477 \[ |
476 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
478 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
813 It is not hard to see that $\cT$ becomes an $n{-}1$-category. |
815 It is not hard to see that $\cT$ becomes an $n{-}1$-category. |
814 \nn{maybe follows from stuff (not yet written) in previous subsection?} |
816 \nn{maybe follows from stuff (not yet written) in previous subsection?} |
815 |
817 |
816 |
818 |
817 |
819 |
|
820 \subsection{The $n{+}1$-category of sphere modules} |
|
821 |
|
822 Outline: |
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823 \begin{itemize} |
|
824 \item |
|
825 \end{itemize} |
818 |
826 |
819 |
827 |
820 |
828 |
821 \medskip |
829 \medskip |
822 \hrule |
830 \hrule |
836 \item traditional $A_\infty$ 1-cat def implies our def |
844 \item traditional $A_\infty$ 1-cat def implies our def |
837 \item ... and vice-versa (already done in appendix) |
845 \item ... and vice-versa (already done in appendix) |
838 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
846 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
839 \item spell out what difference (if any) Top vs PL vs Smooth makes |
847 \item spell out what difference (if any) Top vs PL vs Smooth makes |
840 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
848 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
841 \item define $n{+}1$-cat of $n$-cats; discuss Morita equivalence |
849 (follows as special case of product formula (product with a point). |
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850 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
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851 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
842 \end{itemize} |
852 \end{itemize} |
843 |
853 |
844 |
854 |