627 %\nn{iterated homotopy?} |
627 %\nn{iterated homotopy?} |
628 and also compatible with composition (gluing) in the sense that |
628 and also compatible with composition (gluing) in the sense that |
629 a diagram like the one in Theorem \ref{thm:CH} commutes. |
629 a diagram like the one in Theorem \ref{thm:CH} commutes. |
630 %\nn{repeat diagram here?} |
630 %\nn{repeat diagram here?} |
631 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
631 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
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632 On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide |
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633 with the one coming from Axiom \ref{axiom:morphisms}. |
632 \end{axiom} |
634 \end{axiom} |
633 |
635 |
634 We should strengthen the above $A_\infty$ axiom to apply to families of collar maps. |
636 We should strengthen the above $A_\infty$ axiom to apply to families of collar maps. |
635 To do this we need to explain how collar maps form a topological space. |
637 To do this we need to explain how collar maps form a topological space. |
636 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
638 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
637 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
639 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
638 Having chains on the space of collar maps act gives rise to coherence maps involving |
640 Having chains on the space of collar maps act gives rise to coherence maps involving |
639 weak identities. |
641 weak identities. |
640 We will not pursue this in detail here. |
642 We will not pursue this in detail here. |
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643 |
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644 A variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
641 |
645 |
642 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
646 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
643 into a plain $n$-category (enriched over graded groups). |
647 into a plain $n$-category (enriched over graded groups). |
644 In a different direction, if we enrich over topological spaces instead of chain complexes, |
648 In a different direction, if we enrich over topological spaces instead of chain complexes, |
645 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
649 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |