--- 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.