--- a/text/A-infty.tex Fri Jun 05 23:02:55 2009 +0000
+++ b/text/A-infty.tex Sun Jun 07 00:51:00 2009 +0000
@@ -10,7 +10,6 @@
\subsection{Topological $A_\infty$ categories}
In this section we define a notion of `topological $A_\infty$ category' and sketch an equivalence with the usual definition of $A_\infty$ category. We then define `topological $A_\infty$ modules', and their morphisms and tensor products.
-\paragraph{Categories}
\begin{defn}
\label{defn:topological-Ainfty-category}%
A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$, and for each interval $J$ and objects $a,b \in \Obj(\cC)$, a chain complex $\cC(J;a,b)$, along with
@@ -27,8 +26,15 @@
\end{itemize}
\end{defn}
-\paragraph{Modules}
-We now define left-modules, right-modules and bimodules over a topological $A_\infty category$. We'll say that a right-marked interval is a pair $(J,p)$, diffeomorphic to the pair $([0,1],1)$, and similarly for a left-marked interval. Recall in what follows that when we write a union of interval $J \cup J'$, we're implicitly assuming that both intervals are oriented, and that the union glues together the `highest' point of $J$ with the `lowest' point of $J'$.
+Appendix \ref{sec:comparing-A-infty} explains the translation between this definition and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.)
+
+The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition
+\begin{align*}
+\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.
+
+We now define left-modules, right-modules and bimodules over a topological $A_\infty$ category. We'll say that a right-marked interval is a pair $(J,p)$, diffeomorphic to the pair $([0,1],1)$, and similarly for a left-marked interval. Recall in what follows that when we write a union of interval $J \cup J'$, we're implicitly assuming that both intervals are oriented, and that the union glues together the `highest' point of $J$ with the `lowest' point of $J'$.
\begin{defn}
\label{defn:topological-Ainfty-module}%
@@ -56,10 +62,15 @@
satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}.
\end{defn}
+We now define the tensor product of a left module with a right module. The notion of the self-tensor product of a bimodule is a minor variation which we'll leave to the reader.
+Our definition requires choosing a `fixed' interval, and for simplicity we'll use $[0,1]$, but you should note that the definition is equivariant with respect to diffeomorphisms of this interval.
-\paragraph{Morphisms}
+\begin{defn}
+The tensor product of a left module $\cM$ and a right module $\cN$ over a topological $A_\infty$ category $\cC$, denoted $\cM \tensor_{\cC} \cN$, is a vector space, which we'll specify as the limit of a certain commutative diagram. This (infinite) diagram has vertices indexed by partitions $$[0,1] = [0,x_1] \cup \cdots \cup [x_k,1]$$ and boundary conditions $$a_1, \ldots, a_k \in \Obj(\cC),$$ and arrows labeled by refinements. At each vertex put the vector space $$\cM([0,x_1],0; a_1) \tensor \cC([x_1,x_2];a_1,a_2]) \tensor \cdots \tensor \cC([x_{k-1},x_k];a_{k-1},a_k) \tensor \cN([x_k,1],1;a_k),$$ and on each arrow the corresponding gluing map. Faces of this diagram commute because the gluing maps compose associatively.
+\end{defn}
-\paragraph{Tensor products}
+\newcommand{\lmod}[1]{{}_{#1}{\operatorname{mod}}}
+For completeness, we still need to define morphisms between modules and duals of modules, and explain how the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic. We don't actually need this for the present version of the paper, so the half-written discussion has been banished to Appendix \ref{sec:A-infty-hom-and-duals}.
\subsection{Homological systems of fields}
A homological system of fields $\cF$ is nothing more than a system of fields in the category $\Kom$ of complexes of vector spaces; that is, the set of top level fields with given boundary conditions is always a complex.
@@ -172,14 +183,3 @@
We remark that if $\cF$ takes values in vector spaces, not chain complexes, then the $d_v$ terms vanish, and this coincides with our earlier definition of blob homomology for (non-homological) systems of fields.
\todo{We'll quickly check $d^2=0$.}
-
-
-\subsection{An intermediate gluing theorem}
-
-\begin{thm}[Gluing, intermediate form]
-Suppose $M = M_1 \cup_Y M_2$ is the union of two submanifolds $M_1$ and $M_2$ along a codimension $1$ manifold $Y$. The blob homology of $M$ can be computed as
-\begin{equation*}
-\cB_*(M, \cF) = \cB_*(([0,1], \{0\}, \{1\}), (\cB_*(Y, \cF), \cB_*(M_1, \cF), \cB_*(M_2, \cF))).
-\end{equation*}
-The right hand side is the blob homology of the interval, using ...
-\end{thm}
\ No newline at end of file