Binary file RefereeReport.pdf has changed
--- a/text/kw_macros.tex Wed Aug 10 13:30:17 2011 -0600
+++ b/text/kw_macros.tex Wed Aug 10 15:31:45 2011 -0700
@@ -64,7 +64,7 @@
% \DeclareMathOperator{\pr}{pr} etc.
\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone};
+\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr};
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\hocolim}{hocolim}
--- a/text/ncat.tex Wed Aug 10 13:30:17 2011 -0600
+++ b/text/ncat.tex Wed Aug 10 15:31:45 2011 -0700
@@ -214,12 +214,14 @@
with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$.
Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$.
-We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". When the gluing map is surjective every such element is splittable.
+We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$".
+When the gluing map is surjective every such element is splittable.
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$
as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$.
-We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$
+We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition
+$$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$
a {\it restriction} map and write $\res_{B_i}(a)$
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$.
More generally, we also include under the rubric ``restriction map"
@@ -227,9 +229,14 @@
another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition
of restriction maps.
In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$
-($i = 1, 2$, notation from previous paragraph).
+defined as the composition of the boundary with the first restriction map described above:
+$$
+\cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i)
+.$$
These restriction maps can be thought of as
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$.
+These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$,
+and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$.
Next we consider composition of morphisms.