# HG changeset patch # User scott@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1215400811 0 # Node ID 2f677e283c2653aad90012977a4eee5be51f2382 # Parent f5e553fbd69318d9fcc681baafb4a8581fca5c99 adding some things to the bibliography diff -r f5e553fbd693 -r 2f677e283c26 bibliography/bibliography.bib --- 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, diff -r f5e553fbd693 -r 2f677e283c26 blob1.tex --- 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. diff -r f5e553fbd693 -r 2f677e283c26 preamble.tex --- 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]{} diff -r f5e553fbd693 -r 2f677e283c26 text/article_preamble.tex --- 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