--- a/text/ncat.tex Wed Jun 29 16:17:53 2011 -0700
+++ b/text/ncat.tex Wed Jun 29 16:21:11 2011 -0700
@@ -206,8 +206,8 @@
We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples
we are trying to axiomatize.
If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is
-in the image of the gluing map precisely which the cell complex is in general position
-with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective
+in the image of the gluing map precisely when the cell complex is in general position
+with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective.
If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union
of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified
@@ -999,7 +999,7 @@
\item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions
and collar maps (Axiom \ref{axiom:families}).
\end{itemize}
-The above data must satisfy the following conditions:
+The above data must satisfy the following conditions.
\begin{itemize}
\item The gluing maps are compatible with actions of homeomorphisms and boundary
restrictions (Axiom \ref{axiom:composition}).
@@ -2410,7 +2410,7 @@
This will allow us to define $\cS(X; c)$ independently of the choice of $E$.
First we must define ``inner product", ``non-degenerate" and ``compatible".
-Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image.
+Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ its mirror image.
(We assume we are working in the unoriented category.)
Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$
along their common boundary.