# HG changeset patch # User Scott Morrison # Date 1285367575 25200 # Node ID b138ee4a5938c599327e3c7bf58125342cd62e0b # Parent 62a402dd3e6eb03589f6284e1292fd9e209fa6ed friday afternoon diff -r 62a402dd3e6e -r b138ee4a5938 blob1.tex --- 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} diff -r 62a402dd3e6e -r b138ee4a5938 sandbox.tex --- 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} diff -r 62a402dd3e6e -r b138ee4a5938 text/appendixes/moam.tex --- 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 $j$ diff -r 62a402dd3e6e -r b138ee4a5938 text/basic_properties.tex --- 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 $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. diff -r 62a402dd3e6e -r b138ee4a5938 text/evmap.tex --- 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*} diff -r 62a402dd3e6e -r b138ee4a5938 text/ncat.tex --- 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