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.

   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.

   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.