
diff -r ada83e7228eb -r 7b0a43bdd3c4 blob1.tex
--- a/blob1.tex	Tue Jul 01 01:53:15 2008 +0000
+++ b/blob1.tex	Tue Jul 01 04:00:22 2008 +0000
@@ -229,7 +229,7 @@
 \nn{maybe should look for better name; but this is the name I use elsewhere}
 is a collection of functors $\cC$ from manifolds of dimension $n$ or less
 to sets.
-These functors must satisfy various properties (see KW TQFT notes for details).
+These functors must satisfy various properties (see \cite{kw:tqft} for details).
 For example:
 there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;
 there is a restriction map $\cC(X) \to \cC(\bd X)$;
@@ -381,7 +381,7 @@
 \nn{Roughly, these are (1) the local relations imply (extended) isotopy;
 (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and
 (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$).
-See KW TQFT notes for details.  Need to transfer details to here.}
+See \cite{kw:tqft} for details.  Need to transfer details to here.}
 
 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
@@ -403,7 +403,7 @@
 the $n$-manifold $Y$ modulo local relations.
 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
 is defined to be the dual of $A(Y; c)$.
-(See KW TQFT notes or xxxx for details.)
+(See \cite{kw:tqft} or xxxx for details.)
 
 The blob complex is in some sense the derived version of $A(Y; c)$.
 
@@ -918,15 +918,111 @@
 \subsection{`Topological' $A_\infty$ $n$-categories}
 \label{sec:topological-A-infty}%
 
-This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$
-$n$-category}. The main result of this section is
+This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}.
+The main result of this section is
 
 \begin{thm}
-Topological $A_\infty$ $1$-categories are equivalent to `standard'
-$A_\infty$ $1$-categories.
+Topological $A_\infty$-$1$-categories are equivalent to `standard'
+$A_\infty$-$1$-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
+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?}
 
+\begin{defn}
+\label{defn:topological-algebra}%
+A ``topological $A_\infty$-algebra'' $A$ consists of the 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 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 whenever $\bdy J \cap \bdy J'$ is a single point, a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$,
+\end{enumerate}
+satisfying the following conditions.
+\begin{itemize}
+\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same.
+\item The evaluation chain map is associative, in that the diagram
+\begin{equation*}
+\xymatrix{
+\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} &
+\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\
+\CD{J} \tensor A(J) \ar[r]_{\ev_J} &
+A(J)
+}
+\end{equation*}
+commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.)
+\item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram
+\begin{equation*}
+\xymatrix{
+A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} &&
+A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\
+A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} &&
+A(J \cup J' \cup J'')
+}
+\end{equation*}
+commutes.
+\end{itemize}
+\end{defn}
+
+\begin{rem}
+Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of
+intervals and diffeomorphisms between them to the category of complexes of vector spaces.
+Further, one can combine the second and third pieces of data, asking instead for a map
+\begin{equation*}
+\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J').
+\end{equation*}
+(Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of
+diffeomorphisms in $\CD{J'}$.)
+\end{rem}
+
+To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each
+interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up:
+\begin{equation*}
+\gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+).
+\end{equation*}
+The action of diffeomorphisms, and $k$-parameter families of diffeomorphisms, ignore the boundary conditions.
+
+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 $A$ consists of the data
+\begin{enumerate}
+\item a functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a boundary, to complexes of vector spaces,
+\item along with an `evaluation' map $\ev_K : \CD{K} \tensor M(K) \to M(K)$
+\item whenever $\bdy J \cap K$ is a single point, and isn't the marked point of $K$ \todo{ugh, that's so gross}, a gluing map
+$\gl_{J,K} : A(J) \tensor M(K) \to M(J \cup K)$
+\end{enumerate}
+satisfying the obvious analogous conditions as in Definition \ref{defn:topological-algebra}.
+\end{defn}
+
+\todo{Bimodules, and gluing}
+
+\todo{the motivating example $C_*(\maps(X, M))$}
+
+\todo{higher $n$}
+
+
+\newcommand{\skel}[1]{\operatorname{skeleton}(#1)}
+
+Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your
+favorite diffeomorphism $\phi: I \cup I \to I$.
+\begin{defn}
+We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by
+\begin{equation*}
+m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I).
+\end{equation*}
+Next, we define all the `higher associators' $m_k$ by
+\todo{}
+\end{defn}
+
+Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should
+think of this at the generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition
+in the case the $A$ is actually an associative category.
+\begin{defn}
+\end{defn}
 
 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG
 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty
@@ -1196,8 +1292,8 @@
 \input{text/explicit.tex}
 
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+%\newcommand{\urlprefix}{}
+\bibliographystyle{plain}
 %Included for winedt:
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