compatibility of first and last n-cat axioms; mention stricter variant of last axiom
--- a/text/ncat.tex Mon Dec 27 11:29:54 2010 -0800
+++ b/text/ncat.tex Thu Jan 06 22:47:06 2011 -0800
@@ -629,6 +629,8 @@
a diagram like the one in Theorem \ref{thm:CH} commutes.
%\nn{repeat diagram here?}
%\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}
+On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide
+with the one coming from Axiom \ref{axiom:morphisms}.
\end{axiom}
We should strengthen the above $A_\infty$ axiom to apply to families of collar maps.
@@ -639,6 +641,8 @@
weak identities.
We will not pursue this in detail here.
+A variant on the above axiom would be to drop the ``up to homotopy" and require a strictly associative action.
+
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
into a plain $n$-category (enriched over graded groups).
In a different direction, if we enrich over topological spaces instead of chain complexes,