--- a/sandbox.tex Mon Jul 19 12:26:59 2010 -0700
+++ b/sandbox.tex Mon Jul 19 12:27:19 2010 -0700
@@ -11,26 +11,6 @@
\title{Sandbox}
\begin{document}
-$$
-\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
-\draw (0,0) node(R) {}
- -- (0.75,0) node[below] {$\bar{B}$}
- --(1.5,0) node[circle,fill=black,inner sep=2pt] {}
- arc (0:80:1.5) node[above] {$D \times I$}
- arc (80:180:1.5);
-\foreach \r in {0.3, 0.6, 0.9, 1.2} {
- \draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r);
-}
-\draw[fill=white]
- (R) node[circle,fill=black,inner sep=2pt] {}
- arc (45:65:3) node[below] {$B$}
- arc (65:90:3) node[below] {$A$}
- arc (90:135:3) node[circle,fill=black,inner sep=2pt] {}
- arc (-135:-90:3) node[below] {$C$}
- arc (-90:-45:3);
-\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {$D$};
-\node at (-2,0) {\scalebox{2.0}{$\uparrow f$}};
-\node at (0.2,0.8) {\scalebox{2.0}{$\uparrow \psi$}};
-\end{tikzpicture}
-$$
+
+
\end{document}
--- a/text/ncat.tex Mon Jul 19 12:26:59 2010 -0700
+++ b/text/ncat.tex Mon Jul 19 12:27:19 2010 -0700
@@ -821,8 +821,7 @@
This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
Notice that with $F$ a point, the above example is a construction turning a topological
-$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
-\nn{do we use this notation elsewhere (anymore)?}
+$n$-category $\cC$ into an $A_\infty$ $n$-category.
We think of this as providing a ``free resolution"
of the topological $n$-category.
\nn{say something about cofibrant replacements?}
@@ -1025,8 +1024,6 @@
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit
is more involved.
-\nn{should change to less strange terminology: ``filtration" to ``simplex"
-(search for all occurrences of ``filtration")}
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.
Such sequences (for all $m$) form a simplicial set in $\cell(W)$.
Define $\cl{\cC}(W)$ as a vector space via
@@ -1034,9 +1031,6 @@
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
-(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$,
-the complex $U[m]$ is concentrated in degree $m$.)
-\nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?}
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
summands plus another term using the differential of the simplicial set of $m$-sequences.
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
@@ -1051,14 +1045,13 @@
%combine only two balls at a time; for $n=1$ this version will lead to usual definition
%of $A_\infty$ category}
-We will call $m$ the filtration degree of the complex.
-\nn{is there a more standard term for this?}
+We will call $m$ the simplex degree of the complex.
We can think of this construction as starting with a disjoint copy of a complex for each
-permissible decomposition (filtration degree 0).
+permissible decomposition (simplex degree 0).
Then we glue these together with mapping cylinders coming from gluing maps
-(filtration degree 1).
+(simplex degree 1).
Then we kill the extra homology we just introduced with mapping
-cylinders between the mapping cylinders (filtration degree 2), and so on.
+cylinders between the mapping cylinders (simplex degree 2), and so on.
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
@@ -2220,9 +2213,80 @@
across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$.
(See Figure \ref{jun23d}.)
\begin{figure}[t]
-\begin{equation*}
-\mathfig{.9}{tempkw/jun23d}
-\end{equation*}
+\begin{tikzpicture}
+\node(L) {
+\scalebox{0.5}{
+\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
+\draw[red] (0.75,0) -- +(2,0);
+\draw[red] (0,0) node(R) {}
+ -- (0.75,0) node[below] {}
+ --(1.5,0) node[circle,fill=black,inner sep=2pt] {};
+\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {};
+\draw (1.5,0) arc (0:149:1.5);
+\draw[red]
+ (R) node[circle,fill=black,inner sep=2pt] {}
+ arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {};
+\draw[red] (-5.5,0) -- (-4.2,0);
+\draw (R) arc (45:75:3);
+\draw (150:1.5) arc (74:135:3);
+\node at (-2,0) {\scalebox{2.0}{$B_1$}};
+\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}};
+\node at (-4,1.2) {\scalebox{2.0}{$A$}};
+\node at (-4,-1.2) {\scalebox{2.0}{$C$}};
+\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}};
+\end{tikzpicture}
+}
+};
+\node(M) at (5,4) {
+\scalebox{0.5}{
+\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
+\draw[red] (0.75,0) -- +(2,0);
+\draw[red] (0,0) node(R) {}
+ -- (0.75,0) node[below] {}
+ --(1.5,0) node[circle,fill=black,inner sep=2pt] {};
+\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {};
+\draw(1.5,0) arc (0:149:1.5);
+\draw
+ (R) node[circle,fill=black,inner sep=2pt] {}
+ arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {};
+\draw[red] (-5.5,0) -- (-4.2,0);
+\draw[red] (R) arc (45:75:3);
+\draw[red] (150:1.5) arc (74:135:3);
+\node at (-2,0) {\scalebox{2.0}{$B_1$}};
+\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}};
+\node at (-4,1.2) {\scalebox{2.0}{$A$}};
+\node at (-4,-1.2) {\scalebox{2.0}{$C$}};
+\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}};
+\end{tikzpicture}
+}
+};
+\node(R) at (10,0) {
+\scalebox{0.5}{
+\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
+\draw[red] (0.75,0) -- +(2,0);
+\draw (0,0) node(R) {}
+ -- (0.75,0) node[below] {}
+ --(1.5,0) node[circle,fill=black,inner sep=2pt] {};
+\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {};
+\draw[red] (1.5,0) arc (0:149:1.5);
+\draw
+ (R) node[circle,fill=black,inner sep=2pt] {}
+ arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {};
+\draw[red] (-5.5,0) -- (-4.2,0);
+\draw (R) arc (45:75:3);
+\draw[red] (150:1.5) arc (74:135:3);
+\node at (-2,0) {\scalebox{2.0}{$B_1$}};
+\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}};
+\node at (-4,1.2) {\scalebox{2.0}{$A$}};
+\node at (-4,-1.2) {\scalebox{2.0}{$C$}};
+\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}};
+\end{tikzpicture}
+}
+};
+\draw[->] (L) to[out=90,in=225] node[sloped, above] {push $B_1$} (M);
+\draw[->] (M) to[out=-45,in=90] node[sloped, above] {push $B_2$} (R);
+\draw[->] (L) to[out=-35,in=-145] node[sloped, below] {push $B_1 \cup B_2$} (R);
+\end{tikzpicture}
\caption{A movie move}
\label{jun23d}
\end{figure}