--- a/blob1.tex Thu Sep 23 18:10:35 2010 -0700
+++ b/blob1.tex Fri Sep 24 15:32:55 2010 -0700
@@ -32,9 +32,19 @@
is particularly well suited for work with TQFTs.
\end{abstract}
+\hypersetup{
+ colorlinks, linkcolor={black},
+ citecolor={dark-blue}, urlcolor={medium-blue}
+}
\tableofcontents
+\hypersetup{
+ colorlinks, linkcolor={dark-red},
+ citecolor={dark-blue}, urlcolor={medium-blue}
+}
+
+
%\let\stdsection\section
%\renewcommand\section{\newpage\stdsection}
--- a/sandbox.tex Thu Sep 23 18:10:35 2010 -0700
+++ b/sandbox.tex Fri Sep 24 15:32:55 2010 -0700
@@ -13,128 +13,4 @@
\begin{document}
-\begin{figure}[t]
-\begin{center}
-\begin{tikzpicture}
-
-\node(A) at (-4,0) {
-\begin{tikzpicture}[scale=.8, fill=blue!15!white]
-\filldraw[line width=1.5pt] (-.4,1) .. controls +(-1,-.1) and +(-1,0) .. (0,-1)
- .. controls +(1,0) and +(1,-.1) .. (.4,1) -- (.4,3)
- .. controls +(3,-.4) and +(3,0) .. (0,-3)
- .. controls +(-3,0) and +(-3,-.1) .. (-.4,3) -- cycle;
-\node at (0,-2) {$X$};
-\node (W) at (-2.7,-2) {$W$};
-\node (Y1) at (-1.2,3.5) {$Y$};
-\node (Y2) at (1.4,3.5) {$Y$};
-\node[outer sep=2.3] (y1e) at (-.4,2) {};
-\node[outer sep=2.3] (y2e) at (.4,2) {};
-\node (we1) at (-2.2,-1.1) {};
-\node (we2) at (-.6,-.7) {};
-\draw[->] (Y1) -- (y1e);
-\draw[->] (Y2) -- (y2e);
-\draw[->] (W) .. controls +(0,.5) and +(-.5,-.2) .. (we1);
-\draw[->] (W) .. controls +(.5,0) and +(-.2,-.5) .. (we2);
-\end{tikzpicture}
-};
-
-\node(B) at (4,0) {
-\begin{tikzpicture}[scale=.8, fill=blue!15!white]
-\fill (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)
- .. controls +(1,0) and +(1,0) .. (0,1) -- (0,3)
- .. controls +(3,0) and +(3,0) .. (0,-3)
- .. controls +(-3,0) and +(-3,0) .. (0,3) -- cycle;
-\draw[line width=1.5pt] (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1)
- .. controls +(1,0) and +(1,0) .. (0,1);
-\draw[line width=1.5pt] (0,3) .. controls +(3,0) and +(3,0) .. (0,-3)
- .. controls +(-3,0) and +(-3,0) .. (0,3);
-\draw[line width=.5pt, black!65!white] (0,1) -- (0,3);
-\node at (0,-2) {$X\sgl$};
-\node (W) at (2.7,-2) {$W\sgl$};
-\node (we1) at (2.2,-1.1) {};
-\node (we2) at (.6,-.7) {};
-\draw[->] (W) .. controls +(0,.5) and +(.5,-.2) .. (we1);
-\draw[->] (W) .. controls +(-.5,0) and +(.2,-.5) .. (we2);
-\end{tikzpicture}
-};
-
-
-\draw[->, red!80!green, line width=2pt] (A) -- node[above, black] {glue} (B);
-
-\end{tikzpicture}
-\end{center}
-\caption{Gluing with corners}
-\label{fig:gluing-with-corners}
-\end{figure}
-
-
-
-
-
-blah
-
-\vfill\eject
-
-
-
-\begin{tikzpicture}
-\newcommand{\rr}{6}
-\newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}
-
-\node(A) at (0,0) {
-\begin{tikzpicture}
-\node[red,left] at (0,0) {$y$};
-\draw (0,0) \vertex arc (-120:-105:\rr) node[red,below] {$a$} arc(-105:-90:\rr) \vertex node[red,below](x2) {$x$};
-\draw (0,0) \vertex arc (120:105:\rr) node[red,above] {$a$} arc (105:90:\rr) \vertex node[red,above](x1) {$x$} -- (x2);
-\begin{scope}
- \path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr);
- \foreach \x in {0,0.24,...,3} {
- \draw[green!50!brown] (\x,1) -- (\x,-1);
- }
-\end{scope}
-\draw[red, decorate,decoration={brace,amplitude=5pt}] ($(x1)+(0.2,-0.2)$) -- ($(x2)+(0.2,0.2)$) node[midway, xshift=0.7cm] {$x \times I$};
-\end{tikzpicture}
-};
-
-\node(B) at (-4,-4) {
-\begin{tikzpicture}
-\node[red,left] at (0,0) {$y$};
-\draw (0,0) \vertex
- arc (120:105:\rr) node[red,above] {$a$}
- arc (105:90:\rr) node[red,above] {$x$} \vertex
- arc (90:75:\rr) node[red,above] {$x \times I$}
- arc (75:60:\rr) \vertex node[red,right] {$x$}
- arc (-60:-90:\rr) node[red,below] {$a$}
- arc (-90:-120:\rr);
-\begin{scope}
- \path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr);
- \foreach \x in {0,0.48,...,9} {
- \draw[green!50!brown] (\x/4,1) -- (\x,-1);
- }
-\end{scope}
-\end{tikzpicture}
-};
-
-\node(C) at (4,-4) {
-\begin{tikzpicture}[y=-1cm]
-\node[red,left] at (0,0) {$y$};
-\draw (0,0) \vertex
- arc (120:105:\rr) node[red,below] {$a$}
- arc (105:90:\rr) node[red,below] {$x$} \vertex
- arc (90:75:\rr) node[red,below] {$x \times I$}
- arc (75:60:\rr) \vertex node[red,right] {$x$}
- arc (-60:-90:\rr) node[red,above] {$a$}
- arc (-90:-120:\rr);
-\begin{scope}
- \path[clip] (0,0) arc (-120:-60:\rr) arc (60:120:\rr);
- \foreach \x in {0,0.48,...,9} {
- \draw[green!50!brown] (\x/4,1) -- (\x,-1);
- }
-\end{scope}
-\end{tikzpicture}
-};
-
-\draw[->] (A) -- (B);
-\draw[->] (A) -- (C);
-\end{tikzpicture}
\end{document}
--- a/text/appendixes/moam.tex Thu Sep 23 18:10:35 2010 -0700
+++ b/text/appendixes/moam.tex Fri Sep 24 15:32:55 2010 -0700
@@ -32,7 +32,7 @@
\begin{proof}
(Sketch)
-This is a standard result; see, for example, \nn{need citations: Spanier}.
+This is a standard result; see, for example, \cite[Chapter 4]{MR0210112}.
We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively.
Choose $f(x_{0j})\in D^{0j}_0$ for all $j$
--- a/text/basic_properties.tex Thu Sep 23 18:10:35 2010 -0700
+++ b/text/basic_properties.tex Fri Sep 24 15:32:55 2010 -0700
@@ -3,12 +3,12 @@
\subsection{Basic properties}
\label{sec:basic-properties}
-In this section we complete the proofs of Properties 2-4. \nn{fix these numbers}
-Throughout the paper, where possible, we prove results using Properties 1-4,
+In this section we complete the proofs of Properties \ref{property:disjoint-union}--\ref{property:contractibility}.
+Throughout the paper, where possible, we prove results using Properties \ref{property:functoriality}--\ref{property:contractibility},
rather than the actual definition of blob homology.
This allows the possibility of future improvements on or alternatives to our definition.
In fact, we hope that there may be a characterization of the blob complex in
-terms of Properties 1-4, but at this point we are unaware of one.
+terms of Properties \ref{property:functoriality}--\ref{property:contractibility}, but at this point we are unaware of one.
Recall Property \ref{property:disjoint-union},
that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
--- a/text/evmap.tex Thu Sep 23 18:10:35 2010 -0700
+++ b/text/evmap.tex Fri Sep 24 15:32:55 2010 -0700
@@ -223,7 +223,6 @@
\item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous,
where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on
$\bc_0(B)$ comes from the generating set $\BD_0(B)$.
-\nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
\end{itemize}
We can summarize the above by saying that in the typical continuous family
@@ -277,7 +276,7 @@
whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$.
Now define, for $y\in \btc_{0j}$,
\[
- h(y) = e(y - r(y)) + c(r(y)) .
+ h(y) = e(y - r(y)) - c(r(y)) .
\]
We must now verify that $h$ does the job it was intended to do.
@@ -290,22 +289,21 @@
\end{align*}
For $x\in \btc_{1j}$ we have
\begin{align*}
- \bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) + c(r(\bd_b x)) - e(\bd_t x) && \\
+ \bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) - c(r(\bd_b x)) - e(\bd_t x) && \\
&= \bd_b(e(x)) + e(\bd_b x) && \text{(since $r(\bd_b x) = 0$)} \\
&= x . &&
\end{align*}
For $x\in \btc_{0j}$ with $j\ge 1$ we have
\begin{align*}
- \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) - \bd_t(c(r(x))) +
- e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\
- &= x - r(x) - \bd_t(c(r(x))) + c(r(\bd_t x)) \\
+ \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) + \bd_t(c(r(x))) +
+ e(\bd_t x - r(\bd_t x)) - c(r(\bd_t x)) \\
+ &= x - r(x) + \bd_t(c(r(x))) - c(r(\bd_t x)) \\
&= x - r(x) + r(x) \\
&= x.
\end{align*}
Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram,
-as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$
-\nn{explain why this is true?}
-and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
+as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$
+and $\bd_t(c(r(x))) - c(r(\bd_t x)) = r(x)$.
For $x\in \btc_{00}$ we have
\begin{align*}
--- a/text/ncat.tex Thu Sep 23 18:10:35 2010 -0700
+++ b/text/ncat.tex Fri Sep 24 15:32:55 2010 -0700
@@ -45,7 +45,11 @@
By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
We {\it do not} assume that it is equipped with a
-preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. \nn{List the axiom numbers here, mentioning alternate versions, and also the same in the module section.}
+preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
+
+The axioms for an $n$-category are spread throughout this section.
+Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
+
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on
the boundary), we want a corresponding
@@ -218,6 +222,7 @@
one general type of composition which can be in any ``direction".
\begin{axiom}[Composition]
+\label{axiom:composition}
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).
Let $E = \bd Y$, which is a $k{-}2$-sphere.
@@ -467,6 +472,7 @@
%\addtocounter{axiom}{-1}
\begin{axiom}[Product (identity) morphisms]
+\label{axiom:product}
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
there is a map $\pi^*:\cC(X)\to \cC(E)$.
These maps must satisfy the following conditions.
@@ -612,6 +618,7 @@
%\addtocounter{axiom}{-1}
\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
+\label{axiom:families}
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
\[
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
@@ -1831,7 +1838,7 @@
where $B^j$ is the standard $j$-ball.
A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either
(a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.
-(See Figure \nn{need figure, and improve caption on other figure}.)
+(See Figure \ref{subdividing1marked}.)
We now proceed as in the above module definitions.
\begin{figure}[t] \centering
@@ -1849,6 +1856,41 @@
\label{feb21d}
\end{figure}
+\begin{figure}[t] \centering
+\begin{tikzpicture}[baseline,line width = 2pt]
+\draw[blue][fill=blue!15!white] (0,0) circle (2);
+\fill[red] (0,0) circle (0.1);
+\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} {
+ \draw[red] (0,0) -- (\qm:2);
+% \path (\qa:1) node {\color{green!50!brown} $\cA_\n$};
+% \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$};
+% \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3);
+}
+
+
+\begin{scope}[black, thin]
+\clip (0,0) circle (2);
+\draw (0:1) -- (90:1) -- (180:1) -- (270:1) -- cycle;
+\draw (90:1) -- (90:2.1);
+\draw (180:1) -- (180:2.1);
+\draw (270:1) -- (270:2.1);
+\draw (0:1) -- (15:2.1);
+\draw (0:1) -- (315:1.5) -- (270:1);
+\draw (315:1.5) -- (315:2.1);
+\end{scope}
+
+\node(0marked) at (2.5,2.25) {$0$-marked ball};
+\node(1marked) at (3.5,1) {$1$-marked ball};
+\node(plain) at (3,-1) {plain ball};
+\draw[line width=1pt, green!50!brown, ->] (0marked.270) to[out=270,in=45] (50:1.1);
+\draw[line width=1pt, green!50!brown, ->] (1marked.225) to[out=270,in=45] (0.4,0.1);
+\draw[line width=1pt, green!50!brown, ->] (plain.90) to[out=135,in=45] (-45:1);
+
+\end{tikzpicture}
+\caption{Subdividing a $1$-marked ball into plain, $0$-marked and $1$-marked balls.}
+\label{subdividing1marked}
+\end{figure}
+
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with
\[
\cD(X) \deq \cM(X\times C(S)) .
@@ -2213,8 +2255,7 @@
For $n=1$ we have to check an additional ``global" relations corresponding to
rotating the 0-sphere $E$ around the 1-sphere $\bd X$.
But if $n=1$, then we are in the case of ordinary algebroids and bimodules,
-and this is just the well-known ``Frobenius reciprocity" result for bimodules.
-\nn{find citation for this. Evans and Kawahigashi? Bisch!}
+and this is just the well-known ``Frobenius reciprocity" result for bimodules \cite{MR1424954}.
\medskip