
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.
 




