
diff -r d5caffd01b72 -r 61541264d4b3 text/ncat.tex
--- 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 $v_i$;




