Binary file RefereeReport.pdf has changed

Binary file diagrams/ncat/boundary-collar.pdf has changed

--- a/text/basic_properties.tex	Thu Aug 11 13:26:00 2011 -0700
+++ b/text/basic_properties.tex	Thu Aug 11 13:54:38 2011 -0700
@@ -90,7 +90,7 @@
 $r$ be the restriction of $b$ to $X\setminus S$.
 Note that $S$ is a disjoint union of balls.
 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$.
-Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$.
+Note that if a diagram $b'$ is part of $\bd b$ then $T(b') \sub T(b)$.
 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}), 
 so $f$ and the identity map are homotopic.
 \end{proof}

--- a/text/blobdef.tex	Thu Aug 11 13:26:00 2011 -0700
+++ b/text/blobdef.tex	Thu Aug 11 13:54:38 2011 -0700
@@ -156,7 +156,7 @@
 \end{itemize}
 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not
 a manifold.
-Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs.
+Thus we will need to be more careful when speaking of a field $r$ on the complement of the blobs.
 
 \begin{example} \label{sin1x-example}
 Consider the four subsets of $\Real^3$,
@@ -208,7 +208,7 @@
 %and the entire configuration should be compatible with some gluing decomposition of $X$.
 \begin{defn}
 \label{defn:configuration}
-A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ 
+A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots, B_k\}$ 
 of $X$ such that there exists a gluing decomposition $M_0  \to \cdots \to M_m = X$ of $X$ and 
 for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of 
 $M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$. 
@@ -238,7 +238,7 @@
 \label{defn:blob-diagram}
 A $k$-blob diagram on $X$ consists of
 \begin{itemize}
-\item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$,
+\item a configuration $\{B_1, \ldots, B_k\}$ of $k$ blobs in $X$,
 \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration,
 \end{itemize}
 such that

--- a/text/evmap.tex	Thu Aug 11 13:26:00 2011 -0700
+++ b/text/evmap.tex	Thu Aug 11 13:54:38 2011 -0700
@@ -123,7 +123,7 @@
 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$.
 Let $g$ be the last of the $g_j$'s.
 Choose the sequence $\bar{f}_j$ so that 
-$g(B)$ is contained is an open set of $\cV_1$ and
+$g(B)$ is contained in an open set of $\cV_1$ and
 $g_{j-1}(|f_j|)$ is also contained in an open set of $\cV_1$.
 
 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$
@@ -325,7 +325,7 @@
 \end{proof}
 
 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$}
-if there exists $a'\in \btc_k(S)$
+if there exist $a'\in \btc_k(S)$
 and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$.
 
 \newcommand\sbtc{\btc^{\cU}}
@@ -385,7 +385,7 @@
 Now let $b$ be a generator of $C_2$.
 If $\cU$ is fine enough, there is a disjoint union of balls $V$
 on which $b + h_1(\bd b)$ is supported.
-Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find
+Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_1(X)$, we can find
 $s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}).
 By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find
 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$

--- a/text/hochschild.tex	Thu Aug 11 13:26:00 2011 -0700
+++ b/text/hochschild.tex	Thu Aug 11 13:54:38 2011 -0700
@@ -293,7 +293,7 @@
 $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$.
 Further,
 \begin{align*}
-\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\
+\hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}) b_i\right) \tensor 1 \\
                        & = q - 0
 \end{align*}
 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$).
@@ -341,7 +341,7 @@
 $j$-th term have labelled points $d_{j,i}$ to the left of $*$, $m_j$ at $*$, and $d_{j,i}'$ to the right of $*$,
 and there are labels $c_i$ at the labeled points outside the blob.
 We know that
-$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \tensor \to M),$$
+$$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$
 and so
 \begin{align*}
 \ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\

--- a/text/ncat.tex	Thu Aug 11 13:26:00 2011 -0700
+++ b/text/ncat.tex	Thu Aug 11 13:54:38 2011 -0700
@@ -984,7 +984,7 @@
 There are two differences.
 First, for the $n$-category definition we restrict our attention to balls
 (and their boundaries), while for fields we consider all manifolds.
-Second,  in category definition we directly impose isotopy
+Second,  in the category definition we directly impose isotopy
 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.)
@@ -1239,7 +1239,7 @@
 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
 (Plain colimit, not homotopy colimit.)
 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of 
-the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the 
+the standard ball $B^n$ into $X$, and whose morphisms are given by engulfing some of the 
 embedded balls into a single larger embedded ball.
 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and
 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.
@@ -1488,7 +1488,7 @@
 \end{equation*}
 where $K$ is the vector space spanned by elements $a - g(a)$, with
 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)
-\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.
+\to \psi_{\cC;W,c}(y)$ is the value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.
 
 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit
 is more involved.
@@ -1576,7 +1576,7 @@
 
 Let $z$ be a decomposition of $W$ which is in general position with respect to all of the 
 $x_i$'s and $v_i$'s.
-There there decompositions $x'_i$ and $v'_i$ (for all $i$) such that
+There exist decompositions $x'_i$ and $v'_i$ (for all $i$) such that
 \begin{itemize}
 \item $x'_i$ antirefines to $x_i$ and $z$;
 \item $v'_i$ antirefines to $x'_i$, $x'_{i-1}$ and $v_i$;

--- a/text/tqftreview.tex	Thu Aug 11 13:26:00 2011 -0700
+++ b/text/tqftreview.tex	Thu Aug 11 13:54:38 2011 -0700
@@ -393,7 +393,7 @@
 These motivate the following definition.
 
 \begin{defn}
-A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
+A {\it local relation} is a collection of subspaces $U(B; c) \sub \lf(B; c)$,
 for all $n$-manifolds $B$ which are
 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, 
 satisfying the following properties.