--- a/bibliography/bibliography.bib Mon Jul 07 01:25:14 2008 +0000
+++ b/bibliography/bibliography.bib Mon Jul 07 03:20:11 2008 +0000
@@ -36,6 +36,85 @@
note = {\mathscinet{MR1854636} \arxiv{math.RA/9910179}},
}
+@incollection {MR2061854,
+ AUTHOR = {McClure, James E. and Smith, Jeffrey H.},
+ TITLE = {Operads and cosimplicial objects: an introduction},
+ BOOKTITLE = {Axiomatic, enriched and motivic homotopy theory},
+ SERIES = {NATO Sci. Ser. II Math. Phys. Chem.},
+ VOLUME = {131},
+ PAGES = {133--171},
+ PUBLISHER = {Kluwer Acad. Publ.},
+ ADDRESS = {Dordrecht},
+ YEAR = {2004},
+ MRCLASS = {55P48 (18D50)},
+ MRNUMBER = {MR2061854 (2005b:55018)},
+MRREVIEWER = {David Chataur},
+ note = {\mathscinet{MR2061854} \arxiv{math.QA/0402117}},
+}
+
+@book {MR0420610,
+ AUTHOR = {May, J. P.},
+ TITLE = {The geometry of iterated loop spaces},
+ PUBLISHER = {Springer-Verlag},
+ ADDRESS = {Berlin},
+ YEAR = {1972},
+ PAGES = {viii+175},
+ MRCLASS = {55D35},
+ MRNUMBER = {MR0420610 (54 \#8623b)},
+MRREVIEWER = {J. Stasheff},
+ note = {Lectures Notes in Mathematics, Vol. 271 \mathscinet{MR0420610} \href{http://www.math.uchicago.edu/~may/BOOKS/gils.pdf}{available online}},
+}
+
+@article {MR0236922,
+ AUTHOR = {Boardman, J. M. and Vogt, R. M.},
+ TITLE = {Homotopy-everything {$H$}-spaces},
+ JOURNAL = {Bull. Amer. Math. Soc.},
+ FJOURNAL = {Bulletin of the American Mathematical Society},
+ VOLUME = {74},
+ YEAR = {1968},
+ PAGES = {1117--1122},
+ ISSN = {0002-9904},
+ MRCLASS = {55.42},
+ MRNUMBER = {MR0236922 (38 \#5215)},
+MRREVIEWER = {R. J. Milgram},
+ note = {\mathscinet{MR0236922} \doi{10.1090/S0002-9904-1968-12070-1}},
+}
+
+@book {MR0420609,
+ AUTHOR = {Boardman, J. M. and Vogt, R. M.},
+ TITLE = {Homotopy invariant algebraic structures on topological spaces},
+ SERIES = {Lecture Notes in Mathematics, Vol. 347},
+ PUBLISHER = {Springer-Verlag},
+ ADDRESS = {Berlin},
+ YEAR = {1973},
+ PAGES = {x+257},
+ MRCLASS = {55D35},
+ MRNUMBER = {MR0420609 (54 \#8623a)},
+MRREVIEWER = {J. Stasheff},
+ note = {\mathscinet{MR0420609}},
+}
+
+%The framed little discs operad:
+@article {MR1256989,
+ AUTHOR = {Getzler, E.},
+ TITLE = {Batalin-{V}ilkovisky algebras and two-dimensional topological
+ field theories},
+ JOURNAL = {Comm. Math. Phys.},
+ FJOURNAL = {Communications in Mathematical Physics},
+ VOLUME = {159},
+ YEAR = {1994},
+ NUMBER = {2},
+ PAGES = {265--285},
+ ISSN = {0010-3616},
+ CODEN = {CMPHAY},
+ MRCLASS = {81T70 (17B81 55Q99 58Z05 81T40)},
+ MRNUMBER = {MR1256989 (95h:81099)},
+MRREVIEWER = {J. Stasheff},
+ note = {\mathscinet{MR1256989} \euclid{1104254599}},
+}
+
+
+
@article {MR1917056,
--- a/blob1.tex Mon Jul 07 01:25:14 2008 +0000
+++ b/blob1.tex Mon Jul 07 03:20:11 2008 +0000
@@ -926,24 +926,33 @@
$A_\infty$-$1$-categories.
\end{thm}
-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}
+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
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}
+to [framed] $E_\infty$ algebras
+}
+
+\todo{}
+Various citations we might want to make:
+\begin{itemize}
+\item \cite{MR2061854} McClure and Smith's review article
+\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad)
+\item \cite{MR0236922,MR0420609} Boardman and Vogt
+\item \cite{MR1256989} definition of framed little-discs operad
+\end{itemize}
\begin{defn}
\label{defn:topological-algebra}%
A ``topological $A_\infty$-algebra'' $A$ consists of the following data.
\begin{enumerate}
-\item For each $1$-manifold $J$ diffeomorphic to the standard interval
+\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 For each pair of intervals $J,J'$ an `evaluation' chain map
+\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 $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$.
@@ -963,7 +972,7 @@
A(J'')
}
\end{equation*}
-commutes.
+commutes.
\kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy}
(Here the map $\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$ is a composition: take products of singular chains first, then compose diffeomorphisms.)
%% or the version for separate pieces of data:
@@ -1043,17 +1052,17 @@
The definition of a module follows closely the definition of an algebra or category.
\begin{defn}
\label{defn:topological-module}%
-A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$
+A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$
consists of the following data.
\begin{enumerate}
\item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces.
-\item For each pair of such marked intervals,
+\item For each pair of such marked intervals,
an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$.
\item For each decomposition $K = J\cup K'$ of the marked interval
$K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map
$\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$.
\end{enumerate}
-The above data is required to satisfy
+The above data is required to satisfy
conditions analogous to those in Definition \ref{defn:topological-algebra}.
\end{defn}
@@ -1068,9 +1077,9 @@
There are evaluation maps corresponding to gluing unmarked intervals
to the unmarked ends of $K$ and $L$.
-Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a
+Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a
codimension-0 submanifold of $\bdy X$.
-Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the
+Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the
structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$.
Next we define the coend
@@ -1080,13 +1089,13 @@
\begin{itemize}
\item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N).
-\item For each pair of intervals $N,N'$ an evaluation chain map
+\item For each pair of intervals $N,N'$ an evaluation chain map
$\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$.
\item For each decomposition of intervals $N = K\cup L$,
a gluing map $\gl_{K,L} : M(K,L) \to C(N)$.
\item The evaluation maps are associative.
\nn{up to homotopy?}
-\item Gluing is strictly associative.
+\item Gluing is strictly associative.
That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to
$K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$
agree.
@@ -1097,8 +1106,8 @@
and gluing maps, they factor through the universal thing.
\nn{need to say this in more detail, in particular give the properties of the factoring map}
-Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment
-$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described
+Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment
+$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described
in the above bullet points.
Showing that it is the universal such thing is the content of the gluing theorem proved below.
--- a/preamble.tex Mon Jul 07 01:25:14 2008 +0000
+++ b/preamble.tex Mon Jul 07 03:20:11 2008 +0000
@@ -51,6 +51,7 @@
\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{\euclid}[1]{\href{http://projecteuclid.org/euclid.cmp/#1}{{\tt at Project Euclid: #1}}}
\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}}
@@ -76,7 +77,7 @@
% 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{\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]{}
--- a/text/article_preamble.tex Mon Jul 07 01:25:14 2008 +0000
+++ b/text/article_preamble.tex Mon Jul 07 03:20:11 2008 +0000
@@ -20,10 +20,10 @@
%\marginparwidth 0pt%
%\marginparsep 0pt
-\textwidth 5.5in%
-\textheight 9.0in%
-\oddsidemargin 12pt%
-\evensidemargin 12pt
+%\textwidth 5.5in%
+%\textheight 9.0in%
+%\oddsidemargin 12pt%
+%\evensidemargin 12pt%
-\topmargin -.6in%
-\headsep .5in
+%\topmargin -.6in%
+%\headsep .5in