# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1215288063 0
# Node ID 538f38ddf39502d3094a60cefb2439fd72d86de5
# Parent  e155c518ce31ac01fe9270001be0fbe1d9ab9afe
a few changes to top Ainf cat def, etc.
(checking in frequently)

diff -r e155c518ce31 -r 538f38ddf395 blob1.tex
--- a/blob1.tex	Fri Jul 04 05:22:12 2008 +0000
+++ b/blob1.tex	Sat Jul 05 20:01:03 2008 +0000
@@ -926,26 +926,33 @@
 $A_\infty$-$1$-categories.
 \end{thm}
 
-Before proving this theorem, we embark upon a long string of definitions. For expository purposes, we begin with the $n=1$ special cases, and define
+Before proving this theorem, we embark upon a long string of definitions. 
+\kevin{the \\kevin macro seems to be truncating text of the left side of the page}
+For expository purposes, we begin with the $n=1$ special cases, and define
 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn
 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules.
 \nn{Something about duals?}
 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?}
+\kevin{probably we should say something about the relation
+to [framed] $E_\infty$ algebras}
 
 \begin{defn}
 \label{defn:topological-algebra}%
-A ``topological $A_\infty$-algebra'' $A$ consists of the data
+A ``topological $A_\infty$-algebra'' $A$ consists of the following data.
 \begin{enumerate}
-\item for each $1$-manifold $J$ diffeomorphic to the standard interval $I=\left[0,1\right]$, a complex of vector spaces $A(J)$,
+\item For each $1$-manifold $J$ diffeomorphic to the standard interval 
+$I=\left[0,1\right]$, a complex of vector spaces $A(J)$.
 % either roll functoriality into the evaluation map
-\item and for each pair of intervals $J,J'$ an `evaluation' chain map $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$,
-\item and a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$,
+\item For each pair of intervals $J,J'$ an `evaluation' chain map 
+$\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$.
+\item For each decomposition of intervals $J = J'\cup J''$,
+a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$.
 % or do it as two separate pieces of data
 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$,
 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$,
 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$,
 \end{enumerate}
-satisfying the following conditions.
+This data is required to satisfy the following conditions.
 \begin{itemize}
 \item The evaluation chain map is associative, in that the diagram
 \begin{equation*}
@@ -1018,6 +1025,8 @@
 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)),
 \end{align*}
 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism\todo{inverse, really?!},
+\kevin{I think that's fine.  If we recoil at taking inverses,
+we should use smooth maps instead of diffeos}
 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together.
 \end{enumerate}
 The associativity conditions are trivially satisfied.
diff -r e155c518ce31 -r 538f38ddf395 preamble.tex
--- a/preamble.tex	Fri Jul 04 05:22:12 2008 +0000
+++ b/preamble.tex	Sat Jul 05 20:01:03 2008 +0000
@@ -1,179 +1,179 @@
-%auto-ignore
-%this ensures the arxiv doesn't try to start TeXing here.
-
-\usepackage{amsmath,amssymb,amsfonts}
-\usepackage{ifpdf}
-
-%\ifpdf
-%\usepackage[pdftex,all,color]{xy}
-%\else
-\usepackage[all,color]{xy}
-%\fi
-
-\SelectTips{cm}{}
-% This may speed up compilation of complex documents with many xymatrices.
-%\CompileMatrices
-
-% ----------------------------------------------------------------
-\vfuzz2pt % Don't report over-full v-boxes if over-edge is small
-\hfuzz2pt % Don't report over-full h-boxes if over-edge is small
-% ----------------------------------------------------------------
-
-\def\RCS$#1: #2 ${\expandafter\def\csname RCS#1\endcsname{#2}}
-\RCS$Revision$
-\RCS$Date$
-\newcommand{\versioninfo}{Version \RCSRevision; Last commit \RCSDate}
-
-% diagrams -------------------------------------------------------
-% figures ---------------------------------------------------------
-%%% borrowed from Dror's cobordisms paper, use this to include eps or pdf graphics.
-\ifpdf
-\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/pdf/}
-\else
-\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/eps/}
-\fi
-
-\newcommand{\mathfig}[2]{{\hspace{-3pt}\begin{array}{c}%
-  \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}%
-\end{array}\hspace{-3pt}}}
-\newcommand{\reflectmathfig}[2]{{\hspace{-3pt}\begin{array}{c}%
-  \raisebox{-2.5pt}{\reflectbox{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}}%
-\end{array}\hspace{-3pt}}}
-\newcommand{\rotatemathfig}[3]{{\hspace{-3pt}\begin{array}{c}%
-  \raisebox{-2.5pt}{\rotatebox{#2}{\includegraphics[height=#1\textwidth]{\pathtodiagrams #3}}}%
-\end{array}\hspace{-3pt}}}
-\newcommand{\placefig}[2]{\includegraphics[width=#1\linewidth]{\pathtodiagrams #2}}
-
-\ifpdf
-\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref}
-\else
-\usepackage[dvips,plainpages=false,hypertexnames=false]{hyperref}
-\fi
-\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}}
-\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}}
-\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}}
-
-
-% THEOREMS -------------------------------------------------------
-\theoremstyle{plain}
-\newtheorem*{fact}{Fact}
-\newtheorem{prop}{Proposition}[section]
-\newtheorem{conj}[prop]{Conjecture}
-\newtheorem{thm}[prop]{Theorem}
-\newtheorem{lem}[prop]{Lemma}
-\newtheorem{lemma}[prop]{Lemma}
-\newtheorem{cor}[prop]{Corollary}
-\newtheorem*{cor*}{Corollary}
-\newtheorem*{exc}{Exercise}
-%\theoremstyle{definition}
-\newtheorem{defn}[prop]{Definition}         % numbered definition
-\newtheorem*{defn*}{Definition}             % unnumbered definition
-\newtheorem{question}{Question}
-\newtheorem{property}{Property}
-\newenvironment{rem}{\noindent\textsl{Remark.}}{}  % perhaps looks better than rem above?
-\numberwithin{equation}{section}
-%\numberwithin{figure}{section}
-
-% Marginal notes in draft mode -----------------------------------
-\newcommand{\scott}[1]{\stepcounter{comment}{{\color{blue} $\star^{(\arabic{comment})}$}}\marginpar{\color{blue}  $\star^{(\arabic{comment})}$ \usefont{T1}{scott}{m}{n}  #1 --S}}     % draft mode
-\newcommand{\kevin}[1]{\stepcounter{comment}{\color{red} $\star^{(\arabic{comment})}$}\marginpar{\color{red}  $\star^{(\arabic{comment})}$  #1 --K}}     % draft mode
-\newcommand{\comment}[1]{\stepcounter{comment}$\star^{(\arabic{comment})}$\marginpar{\tiny $\star^{(\arabic{comment})}$ #1}}     % draft mode
-\newcounter{comment}
-\newcommand{\noop}[1]{}
-\newcommand{\todo}[1]{\textbf{TODO: #1}}
-
-% \mathrlap -- a horizontal \smash--------------------------------
-% For comparison, the existing overlap macros:
-% \def\llap#1{\hbox to 0pt{\hss#1}}
-% \def\rlap#1{\hbox to 0pt{#1\hss}}
-\def\clap#1{\hbox to 0pt{\hss#1\hss}}
-\def\mathllap{\mathpalette\mathllapinternal}
-\def\mathrlap{\mathpalette\mathrlapinternal}
-\def\mathclap{\mathpalette\mathclapinternal}
-\def\mathllapinternal#1#2{%
-\llap{$\mathsurround=0pt#1{#2}$}}
-\def\mathrlapinternal#1#2{%
-\rlap{$\mathsurround=0pt#1{#2}$}}
-\def\mathclapinternal#1#2{%
-\clap{$\mathsurround=0pt#1{#2}$}}
-
-% MATH -----------------------------------------------------------
-\newcommand{\Natural}{\mathbb N}
-\newcommand{\Integer}{\mathbb Z}
-\newcommand{\Rational}{\mathbb Q}
-\newcommand{\Real}{\mathbb R}
-\newcommand{\Complex}{\mathbb C}
-\newcommand{\Field}{\mathbb F}
-
-\newcommand{\Id}{\boldsymbol{1}}
-\renewcommand{\imath}{\mathfrak{i}}
-\renewcommand{\jmath}{\mathfrak{j}}
-
-\newcommand{\qRing}{\Integer[q,q^{-1}]}
-\newcommand{\qMod}{\qRing-\operatorname{Mod}}
-\newcommand{\ZMod}{\Integer-\operatorname{Mod}}
-
-\newcommand{\into}{\hookrightarrow}
-\newcommand{\onto}{\twoheadrightarrow}
-\newcommand{\iso}{\cong}
-\newcommand{\actsOn}{\circlearrowright}
-\newcommand{\isoto}{\xrightarrow{\iso}}
-\newcommand{\quismto}{\xrightarrow[\text{q.i.}]{\iso}}
-
-\newcommand{\htpy}{\simeq}
-
-\newcommand{\restrict}[2]{#1{}_{\mid #2}{}}
-\newcommand{\set}[1]{\left\{#1\right\}}
-\newcommand{\setc}[2]{\left\{#1 \;\left| \; #2 \right. \right\}}
-\newcommand{\relations}[2]{\left<#1 \;\left| \; #2 \right. \right>}
-\newcommand{\cone}[3]{C\left(#1 \xrightarrow{#2} #3\right)}
-\newcommand{\pairing}[2]{\left\langle#1 ,#2 \right\rangle}
-
-\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
-\newcommand{\norm}[1]{\left|\left|#1\right|\right|}
-
-\newcommand{\card}[1]{\sharp{#1}}
-
-\newcommand{\bdy}{\partial}
-\newcommand{\compose}{\circ}
-\newcommand{\eset}{\emptyset}
-
-\newcommand{\Cat}{\mathcal{C}}
-
-\newcommand{\psmallmatrix}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)}
-
-\newcommand{\qiq}[2]{[#1]_{#2}}
-\newcommand{\qi}[1]{\qiq{#1}{q}}
-\newcommand{\qdim}{\operatorname{dim_q}}
-
-\newcommand{\directSum}{\oplus}
-\newcommand{\DirectSum}{\bigoplus}
-\newcommand{\tensor}{\otimes}
-\newcommand{\Tensor}{\bigotimes}
-
-\newcommand{\db}[1]{\left(\left(#1\right)\right)}
-
-\newcommand{\Mat}[1]{\mathbf{Mat}\left(#1\right)}
-\newcommand{\Inv}[1]{\operatorname{Inv}\left(#1\right)}
-\newcommand{\Hom}[3]{\operatorname{Hom}_{#1}\left(#2,#3\right)}
-\newcommand{\End}[1]{\operatorname{End}\left(#1\right)}
-
-\newcommand{\CD}[1]{C_*(\Diff(#1))}
-
-\newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}}
-\newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}}
-
-\newcommand{\grading}[1]{{\color{blue}\{#1\}}}
-\newcommand{\shift}[1]{\left[#1\right]}
-
-\newenvironment{narrow}[2]{%
-\vspace{-0.4cm}% horrible hack, by scott % this only seems to be appropriate in beamer mode...
-\begin{list}{}{%
-\setlength{\topsep}{0pt}%
-\setlength{\leftmargin}{#1}%
-\setlength{\rightmargin}{#2}%
-\setlength{\listparindent}{\parindent}%
-\setlength{\itemindent}{\parindent}%
-\setlength{\parsep}{\parskip}}%
-\item[]}{\end{list}}
-% ----------------------------------------------------------------
+%auto-ignore
+%this ensures the arxiv doesn't try to start TeXing here.
+
+\usepackage{amsmath,amssymb,amsfonts}
+\usepackage{ifpdf}
+
+%\ifpdf
+%\usepackage[pdftex,all,color]{xy}
+%\else
+\usepackage[all,color]{xy}
+%\fi
+
+\SelectTips{cm}{}
+% This may speed up compilation of complex documents with many xymatrices.
+%\CompileMatrices
+
+% ----------------------------------------------------------------
+\vfuzz2pt % Don't report over-full v-boxes if over-edge is small
+\hfuzz2pt % Don't report over-full h-boxes if over-edge is small
+% ----------------------------------------------------------------
+
+\def\RCS$#1: #2 ${\expandafter\def\csname RCS#1\endcsname{#2}}
+\RCS$Revision$
+\RCS$Date$
+\newcommand{\versioninfo}{Version \RCSRevision; Last commit \RCSDate}
+
+% diagrams -------------------------------------------------------
+% figures ---------------------------------------------------------
+%%% borrowed from Dror's cobordisms paper, use this to include eps or pdf graphics.
+\ifpdf
+\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/pdf/}
+\else
+\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/eps/}
+\fi
+
+\newcommand{\mathfig}[2]{{\hspace{-3pt}\begin{array}{c}%
+  \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}%
+\end{array}\hspace{-3pt}}}
+\newcommand{\reflectmathfig}[2]{{\hspace{-3pt}\begin{array}{c}%
+  \raisebox{-2.5pt}{\reflectbox{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}}%
+\end{array}\hspace{-3pt}}}
+\newcommand{\rotatemathfig}[3]{{\hspace{-3pt}\begin{array}{c}%
+  \raisebox{-2.5pt}{\rotatebox{#2}{\includegraphics[height=#1\textwidth]{\pathtodiagrams #3}}}%
+\end{array}\hspace{-3pt}}}
+\newcommand{\placefig}[2]{\includegraphics[width=#1\linewidth]{\pathtodiagrams #2}}
+
+\ifpdf
+\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref}
+\else
+\usepackage[dvips,plainpages=false,hypertexnames=false]{hyperref}
+\fi
+\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}}
+\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}}
+\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}}
+
+
+% THEOREMS -------------------------------------------------------
+\theoremstyle{plain}
+\newtheorem*{fact}{Fact}
+\newtheorem{prop}{Proposition}[section]
+\newtheorem{conj}[prop]{Conjecture}
+\newtheorem{thm}[prop]{Theorem}
+\newtheorem{lem}[prop]{Lemma}
+\newtheorem{lemma}[prop]{Lemma}
+\newtheorem{cor}[prop]{Corollary}
+\newtheorem*{cor*}{Corollary}
+\newtheorem*{exc}{Exercise}
+%\theoremstyle{definition}
+\newtheorem{defn}[prop]{Definition}         % numbered definition
+\newtheorem*{defn*}{Definition}             % unnumbered definition
+\newtheorem{question}{Question}
+\newtheorem{property}{Property}
+\newenvironment{rem}{\noindent\textsl{Remark.}}{}  % perhaps looks better than rem above?
+\numberwithin{equation}{section}
+%\numberwithin{figure}{section}
+
+% Marginal notes in draft mode -----------------------------------
+\newcommand{\scott}[1]{\stepcounter{comment}{{\color{blue} $\star^{(\arabic{comment})}$}}\marginpar{\color{blue}  $\star^{(\arabic{comment})}$ \usefont{T1}{scott}{m}{n}  #1 --S}}     % draft mode
+\newcommand{\kevin}[1]{\stepcounter{comment}{\color{green} $\star^{(\arabic{comment})}$}\marginpar{\color{green}  $\star^{(\arabic{comment})}$  #1 --K}}     % draft mode
+\newcommand{\comment}[1]{\stepcounter{comment}$\star^{(\arabic{comment})}$\marginpar{\tiny $\star^{(\arabic{comment})}$ #1}}     % draft mode
+\newcounter{comment}
+\newcommand{\noop}[1]{}
+\newcommand{\todo}[1]{\textbf{TODO: #1}}
+
+% \mathrlap -- a horizontal \smash--------------------------------
+% For comparison, the existing overlap macros:
+% \def\llap#1{\hbox to 0pt{\hss#1}}
+% \def\rlap#1{\hbox to 0pt{#1\hss}}
+\def\clap#1{\hbox to 0pt{\hss#1\hss}}
+\def\mathllap{\mathpalette\mathllapinternal}
+\def\mathrlap{\mathpalette\mathrlapinternal}
+\def\mathclap{\mathpalette\mathclapinternal}
+\def\mathllapinternal#1#2{%
+\llap{$\mathsurround=0pt#1{#2}$}}
+\def\mathrlapinternal#1#2{%
+\rlap{$\mathsurround=0pt#1{#2}$}}
+\def\mathclapinternal#1#2{%
+\clap{$\mathsurround=0pt#1{#2}$}}
+
+% MATH -----------------------------------------------------------
+\newcommand{\Natural}{\mathbb N}
+\newcommand{\Integer}{\mathbb Z}
+\newcommand{\Rational}{\mathbb Q}
+\newcommand{\Real}{\mathbb R}
+\newcommand{\Complex}{\mathbb C}
+\newcommand{\Field}{\mathbb F}
+
+\newcommand{\Id}{\boldsymbol{1}}
+\renewcommand{\imath}{\mathfrak{i}}
+\renewcommand{\jmath}{\mathfrak{j}}
+
+\newcommand{\qRing}{\Integer[q,q^{-1}]}
+\newcommand{\qMod}{\qRing-\operatorname{Mod}}
+\newcommand{\ZMod}{\Integer-\operatorname{Mod}}
+
+\newcommand{\into}{\hookrightarrow}
+\newcommand{\onto}{\twoheadrightarrow}
+\newcommand{\iso}{\cong}
+\newcommand{\actsOn}{\circlearrowright}
+\newcommand{\isoto}{\xrightarrow{\iso}}
+\newcommand{\quismto}{\xrightarrow[\text{q.i.}]{\iso}}
+
+\newcommand{\htpy}{\simeq}
+
+\newcommand{\restrict}[2]{#1{}_{\mid #2}{}}
+\newcommand{\set}[1]{\left\{#1\right\}}
+\newcommand{\setc}[2]{\left\{#1 \;\left| \; #2 \right. \right\}}
+\newcommand{\relations}[2]{\left<#1 \;\left| \; #2 \right. \right>}
+\newcommand{\cone}[3]{C\left(#1 \xrightarrow{#2} #3\right)}
+\newcommand{\pairing}[2]{\left\langle#1 ,#2 \right\rangle}
+
+\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
+\newcommand{\norm}[1]{\left|\left|#1\right|\right|}
+
+\newcommand{\card}[1]{\sharp{#1}}
+
+\newcommand{\bdy}{\partial}
+\newcommand{\compose}{\circ}
+\newcommand{\eset}{\emptyset}
+
+\newcommand{\Cat}{\mathcal{C}}
+
+\newcommand{\psmallmatrix}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)}
+
+\newcommand{\qiq}[2]{[#1]_{#2}}
+\newcommand{\qi}[1]{\qiq{#1}{q}}
+\newcommand{\qdim}{\operatorname{dim_q}}
+
+\newcommand{\directSum}{\oplus}
+\newcommand{\DirectSum}{\bigoplus}
+\newcommand{\tensor}{\otimes}
+\newcommand{\Tensor}{\bigotimes}
+
+\newcommand{\db}[1]{\left(\left(#1\right)\right)}
+
+\newcommand{\Mat}[1]{\mathbf{Mat}\left(#1\right)}
+\newcommand{\Inv}[1]{\operatorname{Inv}\left(#1\right)}
+\newcommand{\Hom}[3]{\operatorname{Hom}_{#1}\left(#2,#3\right)}
+\newcommand{\End}[1]{\operatorname{End}\left(#1\right)}
+
+\newcommand{\CD}[1]{C_*(\Diff(#1))}
+
+\newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}}
+\newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}}
+
+\newcommand{\grading}[1]{{\color{blue}\{#1\}}}
+\newcommand{\shift}[1]{\left[#1\right]}
+
+\newenvironment{narrow}[2]{%
+\vspace{-0.4cm}% horrible hack, by scott % this only seems to be appropriate in beamer mode...
+\begin{list}{}{%
+\setlength{\topsep}{0pt}%
+\setlength{\leftmargin}{#1}%
+\setlength{\rightmargin}{#2}%
+\setlength{\listparindent}{\parindent}%
+\setlength{\itemindent}{\parindent}%
+\setlength{\parsep}{\parskip}}%
+\item[]}{\end{list}}
+% ----------------------------------------------------------------