Binary file RefereeReport.pdf has changed
--- a/blob to-do Tue Oct 04 22:45:08 2011 -0700
+++ b/blob to-do Thu Oct 06 12:55:57 2011 -0700
@@ -42,6 +42,6 @@
====== Scott ======
-* SCOTT will go through appendix C.2 and make it better
+* SCOTT will go through appendix C.2 and make it better (Schulman's example?)
* SCOTT: review/proof-read recent KW changes, especially colimit section and n-cat axioms
--- a/text/a_inf_blob.tex Tue Oct 04 22:45:08 2011 -0700
+++ b/text/a_inf_blob.tex Thu Oct 06 12:55:57 2011 -0700
@@ -2,9 +2,13 @@
\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories}
\label{sec:ainfblob}
-Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the following
anticlimactically tautological definition of the blob
-complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
+complex.
+\begin{defn}
+The blob complex
+ $\bc_*(M;\cC)$ of an $n$-manifold $n$ with coefficients in an $A_\infty$ $n$-category is the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
+\end{defn}
We will show below
in Corollary \ref{cor:new-old}
@@ -335,7 +339,7 @@
\subsection{A gluing theorem}
\label{sec:gluing}
-Next we prove a gluing theorem.
+Next we prove a gluing theorem. Throughout this section fix a particular $n$-dimensional system of fields $\cE$ and local relations. Each blob complex below is with respect to this $\cE$.
Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.
We will need an explicit collar on $Y$, so rewrite this as
$X = X_1\cup (Y\times J) \cup X_2$.
@@ -364,7 +368,7 @@
\begin{thm}
\label{thm:gluing}
-When $k=n$ above, $\bc(X)$ is homotopy equivalent to $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+Suppose $X$ is an $n$-manifold, and $X = X_1\cup (Y\times J) \cup X_2$ (i.e. just as with $k=n$ above). Then $\bc(X)$ is homotopy equivalent to the $A_\infty$ tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
\begin{proof}
--- a/text/appendixes/moam.tex Tue Oct 04 22:45:08 2011 -0700
+++ b/text/appendixes/moam.tex Thu Oct 06 12:55:57 2011 -0700
@@ -2,6 +2,9 @@
\section{The method of acyclic models} \label{sec:moam}
+In this section we recall the method of acyclic models for the reader's convenience. The material presented here is closely modeled on \cite[Chapter 4]{MR0210112}.
+We use this method throughout the paper (c.f. Lemma \ref{support-shrink}, Theorem \ref{thm:product}, Theorem \ref{thm:gluing} and Theorem \ref{thm:map-recon}), as it provides a very convenient way to show the existence of a chain map with desired properties, even when many non-canonical choices are required in order to construct one, and further to show the up-to-homotopy uniqueness of such maps.
+
Let $F_*$ and $G_*$ be chain complexes.
Assume $F_k$ has a basis $\{x_{kj}\}$
(that is, $F_*$ is free and we have specified a basis).
--- a/text/ncat.tex Tue Oct 04 22:45:08 2011 -0700
+++ b/text/ncat.tex Thu Oct 06 12:55:57 2011 -0700
@@ -1143,11 +1143,15 @@
invariance in dimension $n$, while in the fields definition we
instead remember a subspace of local relations which contain differences of isotopic fields.
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
-Thus a \nn{lemma-ize} system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
+Thus
+\begin{lem}
+\label{lem:ncat-from-fields}
+A system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
balls and, at level $n$, quotienting out by the local relations:
\begin{align*}
\cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases}
\end{align*}
+\end{lem}
This $n$-category can be thought of as the local part of the fields.
Conversely, given a disk-like $n$-category we can construct a system of fields via
a colimit construction; see \S \ref{ss:ncat_fields} below.
@@ -1250,6 +1254,8 @@
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$.
\end{example}
+This last example generalizes Lemma \ref{lem:ncat-from-fields} above which produced an $n$-category from an $n$-dimensional system of fields and local relations. Taking $W$ to be the point recovers that statement.
+
The next example is only intended to be illustrative, as we don't specify
which definition of a ``traditional $n$-category" we intend.
Further, most of these definitions don't even have an agreed-upon notion of
@@ -1323,6 +1329,8 @@
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to
homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.
+Instead of using the TQFT invariant $\cA$ as in Example \ref{ex:ncats-from-tqfts} above, we can turn an $n$-dimensional system of fields and local relations into an $A_\infty$ $n$-category using the blob complex. With a codimension $k$ fiber, we obtain an $A_\infty$ $k$-category:
+
\begin{example}[Blob complexes of balls (with a fiber)]
\rm
\label{ex:blob-complexes-of-balls}