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939 (This assumes some prior notion of $A_\infty$ 1-category.) |
939 (This assumes some prior notion of $A_\infty$ 1-category.) |
940 We are not currently aware of any examples which require this sort of greater generality, so we think it best |
940 We are not currently aware of any examples which require this sort of greater generality, so we think it best |
941 to refrain from settling on a preferred version of the axiom until |
941 to refrain from settling on a preferred version of the axiom until |
942 we have a greater variety of examples to guide the choice. |
942 we have a greater variety of examples to guide the choice. |
943 |
943 |
944 \nn{say something about isotopy invariance being a special case} |
944 Note that if we think of an ordinary 1-category as an $A_\infty$ 1-category where $k$-morphisms are identities for $k>1$, |
|
945 then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}. |
945 |
946 |
946 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
947 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
947 In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
948 In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
949 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
949 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
950 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |