--- a/text/appendixes/comparing_defs.tex Sat Dec 10 21:07:44 2011 -0800
+++ b/text/appendixes/comparing_defs.tex Sat Dec 10 21:07:51 2011 -0800
@@ -530,7 +530,7 @@
Figure \ref{fig:horizontal-compositions-equal}illustrates part of the proof that these four 2-morphisms are equal.
Similar arguments show that horizontal composition is associative.
\begin{figure}[t]
-\begin{equation*}
+\begin{align*}
\raisebox{-.9cm}{
\begin{tikzpicture}
\draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0)
@@ -544,7 +544,7 @@
.. controls +(-1,-.8) and +(1,-.8) .. node[below] {$c$} (0,0);
\draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$g$} +(0,.8);
\end{tikzpicture}}
-\;=\;
+\;&=\;
\raisebox{-1.9cm}{
\begin{tikzpicture}
\draw (0,0) coordinate (p1);
@@ -569,11 +569,86 @@
\draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7);
\draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7);
\draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0);
-\end{tikzpicture}}
-\end{equation*}
-\begin{equation*}
-\mathfig{0.6}{triangle/triangle3b}
-\end{equation*}
+\end{tikzpicture}} \\
+\;&=\;
+\raisebox{-2.1cm}{
+\begin{tikzpicture}
+ \draw (0,0) coordinate (p1);
+ \draw (5.8,0) coordinate (p2);
+ \draw (2.9,0) coordinate (pu);
+ \draw (2.9,-.9) coordinate (pd);
+ \begin{scope}
+ \clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu)
+ .. controls +(.5,-.3) and +(-.6,-.3) .. (p2)
+ .. controls +(-.6,-.9) and +(.5,0) .. (pd)
+ .. controls +(-.5,0) and +(.6,-.9) .. (p1);
+ \foreach \t in {0,.03,...,1} {
+ \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4);
+ }
+ \end{scope}
+ \draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu)
+ .. controls +(.5,-.3) and +(-.6,-.3) .. (p2)
+ .. controls +(-.6,-.9) and +(.5,0) .. (pd)
+ .. controls +(-.5,0) and +(.6,-.9) .. (p1);
+ \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu);
+ \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu);
+ \draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7);
+ \draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7);
+ \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5);
+\end{tikzpicture}} \\
+\;&=\;
+\raisebox{-1.9cm}{
+\begin{tikzpicture}[y=-1cm]
+ \draw (0,0) coordinate (p1);
+ \draw (5.8,0) coordinate (p2);
+ \draw (2.9,.3) coordinate (pu);
+ \draw (2.9,-.3) coordinate (pd);
+ \begin{scope}
+ \clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu)
+ .. controls +(.5,0) and +(-.6,.3) .. (p2)
+ .. controls +(-.6,-.3) and +(.5,0) .. (pd)
+ .. controls +(-.5,0) and +(.6,-.3) .. (p1);
+ \foreach \t in {0,.03,...,1} {
+ \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4);
+ }
+ \end{scope}
+ \draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu)
+ .. controls +(.5,0) and +(-.6,.3) .. (p2)
+ .. controls +(-.6,-.3) and +(.5,0) .. (pd)
+ .. controls +(-.5,0) and +(.6,-.3) .. (p1);
+ \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd);
+ \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu);
+ \draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7);
+ \draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7);
+ \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0);
+\end{tikzpicture}} \\
+\;&=\;
+\raisebox{-1.0cm}{
+\begin{tikzpicture}[y=-1cm]
+ \draw (0,0) coordinate (p1);
+ \draw (5.8,0) coordinate (p2);
+ \draw (2.9,0) coordinate (pu);
+ \draw (2.9,-.9) coordinate (pd);
+ \begin{scope}
+ \clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu)
+ .. controls +(.5,-.3) and +(-.6,-.3) .. (p2)
+ .. controls +(-.6,-.9) and +(.5,0) .. (pd)
+ .. controls +(-.5,0) and +(.6,-.9) .. (p1);
+ \foreach \t in {0,.03,...,1} {
+ \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4);
+ }
+ \end{scope}
+ \draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu)
+ .. controls +(.5,-.3) and +(-.6,-.3) .. (p2)
+ .. controls +(-.6,-.9) and +(.5,0) .. (pd)
+ .. controls +(-.5,0) and +(.6,-.9) .. (p1);
+ \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu);
+ \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu);
+ \draw[<-, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7);
+ \draw[<-, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7);
+ \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, above] {$(b\cdot d)\times I$} -- (3.3,-0.5);
+\end{tikzpicture}}
+\end{align*}
\caption{Horizontal composition of 2-morphisms}
\label{fzo5}
\end{figure}
--- a/text/evmap.tex Sat Dec 10 21:07:44 2011 -0800
+++ b/text/evmap.tex Sat Dec 10 21:07:51 2011 -0800
@@ -50,7 +50,7 @@
\medskip
If $b$ is a blob diagram in $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that the {\it support} of $b$, denoted
-$\supp(b)$ or $|b|$, to be the union of the blobs of $b$.
+$\supp(b)$ or $|b|$, is the union of the blobs of $b$.
%For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union
%of the supports of the blob diagrams which appear in it.
More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if
@@ -64,14 +64,14 @@
$f$ is supported on $Y$.
If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism
-(cf. end of \S \ref{ss:syst-o-fields}),
+(cf.\ the end of \S \ref{ss:syst-o-fields}),
we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$.
\medskip
Fix $\cU$, an open cover of $X$.
Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$
-of all blob diagrams in which every blob is contained in some open set of $\cU$,
+generated by blob diagrams such that every blob is contained in some open set of $\cU$,
and moreover each field labeling a region cut out by the blobs is splittable
into fields on smaller regions, each of which is contained in some open set of $\cU$.
@@ -114,7 +114,7 @@
The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$.
Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and
-fine enough that a condition stated later in the proof is satisfied.
+fine enough that a condition stated later in this proof is satisfied.
Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$.
Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions
specified at the end of this paragraph.
@@ -426,7 +426,7 @@
\eq{
e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) ,
}
-well-defined up to (coherent) homotopy,
+well-defined up to coherent homotopy,
such that
\begin{enumerate}
\item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of