# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1312939719 21600
# Node ID 84bb5ab4c85cb9638b74bd6355bff17447cc331a
# Parent  daa522adb4881888b2cd21c74f24c2586c11a978
unfinished edits to fam-o-homeo lemma and EB_n algebra example

diff -r daa522adb488 -r 84bb5ab4c85c notes for response to referee report
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/notes for response to referee report	Tue Aug 09 19:28:39 2011 -0600
@@ -0,0 +1,11 @@
+notes for response to referee report
+
+- We incorporated all the suggestions of the referee, with the following exceptions...
+
+- RR7: ...
+
+
+
+
+
+- (?) include blob_changes_v3 (?)
diff -r daa522adb488 -r 84bb5ab4c85c text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Fri Aug 05 12:27:11 2011 -0600
+++ b/text/appendixes/famodiff.tex	Tue Aug 09 19:28:39 2011 -0600
@@ -231,26 +231,29 @@
 \end{lemma}
 
 \begin{proof}
-We will imitate the proof of Corollary 1.3 of \cite{MR0283802}.
-
-Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$.
-After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$
-such that $g^{-1}\circ f(P)$ is a small neighborhood of the 
-identity in $\Homeo(X)$.
-The sense of ``small" we mean will be explained below.
-It depends only on $\cU$ and some auxiliary covers.
-
-We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.
+The proof is similar to the proof of Corollary 1.3 of \cite{MR0283802}.
 
 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite.
 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$.
 
-We will need some wiggle room, so for each $\alpha$ choose open sets
+We will need some wiggle room, so for each $\alpha$ choose a large finite number of open sets
 \[
-	U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset \cdots \supset U_\alpha^N
+	U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset U_\alpha^2 \supset \cdots
 \]
 so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that
 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$.
+\nn{say specifically how many we need?}
+
+
+Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$.
+After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$
+such that $g^{-1}\circ f(P)$ is contained in a small neighborhood of the 
+identity in $\Homeo(X)$.
+The sense of ``small" we mean will be explained below.
+It depends only on $\cU$ and the choice of $U_\alpha^i$'s.
+
+We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.
+
 
 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us
 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties:
diff -r daa522adb488 -r 84bb5ab4c85c text/ncat.tex
--- a/text/ncat.tex	Fri Aug 05 12:27:11 2011 -0600
+++ b/text/ncat.tex	Tue Aug 09 19:28:39 2011 -0600
@@ -945,8 +945,13 @@
 
 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. 
 In fact, the alternative construction $\btc_*(X)$ of the blob complex described in \S \ref{ss:alt-def} 
-gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; 
-since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.
+gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. 
+%since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.
+For future reference we make the following definition.
+
+\begin{defn}
+A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
+\end{defn}
 
 \noop{
 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
@@ -1220,7 +1225,7 @@
 Let $A$ be an $\cE\cB_n$-algebra.
 Note that this implies a $\Diff(B^n)$ action on $A$, 
 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
-We will define an $A_\infty$ $n$-category $\cC^A$.
+We will define a strict $A_\infty$ $n$-category $\cC^A$.
 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
 In other words, the $k$-morphisms are trivial for $k<n$.
 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
@@ -1237,10 +1242,13 @@
 also comes from the $\cE\cB_n$ action on $A$.
 %\nn{should we spell this out?}
 
-Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to 
 an $\cE\cB_n$-algebra.
 %\nn{The paper is already long; is it worth giving details here?}
+% According to the referee, yes it is...
+Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball.
+\nn{need to finish this}
 
 If we apply the homotopy colimit construction of the next subsection to this example, 
 we get an instance of Lurie's topological chiral homology construction.