930 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
930 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
931 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
931 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
932 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. |
932 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. |
933 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
933 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
934 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
934 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
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935 |
935 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
936 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
936 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
937 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
937 functor $\bbc \to \cS$ of $A_\infty$ 1-categories. |
938 functor $\bbc \to \cS$ of $A_\infty$ 1-categories. |
938 (This assumes some prior notion of $A_\infty$ 1-category.) |
939 (This assumes some prior notion of $A_\infty$ 1-category.) |
939 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 |
940 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 |
941 we have a greater variety of examples to guide the choice. |
942 we have a greater variety of examples to guide the choice. |
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943 |
|
944 \nn{say something about isotopy invariance being a special case} |
942 |
945 |
943 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
946 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
944 In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
947 In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} |
945 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
946 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
949 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |