# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1294382826 28800
# Node ID 72a1d5014abc07c51ca181429ba2042558da4eea
# Parent  e0bd7c5ec86419a30d59bda73b24586a3f32bafe
compatibility of first and last n-cat axioms; mention stricter variant of last axiom

diff -r e0bd7c5ec864 -r 72a1d5014abc text/ncat.tex
--- 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,