Binary file RefereeReport.pdf has changed
--- a/text/a_inf_blob.tex Fri Jul 15 14:48:43 2011 -0700
+++ b/text/a_inf_blob.tex Fri Jul 15 15:03:22 2011 -0700
@@ -119,7 +119,7 @@
the case.
(Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.)
However, we {\it can} find another decomposition $L$ such that $L$ shares common
-refinements with both $K$ and $K'$.
+refinements with both $K$ and $K'$. (For instance, in the example above, $L$ can be the graph of $y=x^2+1$.)
This follows from Axiom \ref{axiom:vcones}, which in turn follows from the
splitting axiom for the system of fields $\cE$.
Let $KL$ and $K'L$ denote these two refinements.
--- a/text/appendixes/comparing_defs.tex Fri Jul 15 14:48:43 2011 -0700
+++ b/text/appendixes/comparing_defs.tex Fri Jul 15 15:03:22 2011 -0700
@@ -13,8 +13,7 @@
One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms.
One should also show that composing the two arrows (between traditional and disk-like $n$-categories)
yields the appropriate sort of equivalence on each side.
-Since we haven't given a definition for functors between disk-like $n$-categories
-(the paper is already too long!), we do not pursue this here.
+Since we haven't given a definition for functors between disk-like $n$-categories, we do not pursue this here.
We emphasize that we are just sketching some of the main ideas in this appendix ---
it falls well short of proving the definitions are equivalent.
--- a/text/deligne.tex Fri Jul 15 14:48:43 2011 -0700
+++ b/text/deligne.tex Fri Jul 15 15:03:22 2011 -0700
@@ -124,7 +124,7 @@
In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries
do not overlap, we can perform them in reverse order or simultaneously.
-There is an operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary
+There is a colored operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary
of one cylinder into one of the inner boundaries of another cylinder.
We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc.
--- a/text/evmap.tex Fri Jul 15 14:48:43 2011 -0700
+++ b/text/evmap.tex Fri Jul 15 15:03:22 2011 -0700
@@ -49,7 +49,7 @@
\medskip
-If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted
+If $b$ is a blob diagram in $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that the {\it support} of $b$, denoted
$\supp(b)$ or $|b|$, to be the union of the blobs of $b$.
%For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union
%of the supports of the blob diagrams which appear in it.
--- a/text/ncat.tex Fri Jul 15 14:48:43 2011 -0700
+++ b/text/ncat.tex Fri Jul 15 15:03:22 2011 -0700
@@ -944,7 +944,7 @@
then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}.
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 of the blob complex described in \S \ref{ss:alt-def}
+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.
@@ -1143,11 +1143,11 @@
For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$.
Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex
\[
- C_*(\Maps_c(X\times F \to T)),
+ C_*(\Maps_c(X \to T)),
\]
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
and $C_*$ denotes singular chains.
-Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$,
+Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$,
we get an $A_\infty$ $n$-category enriched over spaces.
\end{example}