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641 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
641 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
642 Having chains on the space of collar maps act gives rise to coherence maps involving |
642 Having chains on the space of collar maps act gives rise to coherence maps involving |
643 weak identities. |
643 weak identities. |
644 We will not pursue this in detail here. |
644 We will not pursue this in detail here. |
645 |
645 |
646 A variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
646 A potential variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. (In fact, the alternative construction of the blob complex described in \S ?? gives $n$-categories as in Example ?? which satisfy this stronger axiom; since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.) |
647 |
647 |
648 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
648 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
649 into a ordinary $n$-category (enriched over graded groups). |
649 into a ordinary $n$-category (enriched over graded groups). |
650 In a different direction, if we enrich over topological spaces instead of chain complexes, |
650 In a different direction, if we enrich over topological spaces instead of chain complexes, |
651 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
651 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |