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\input{text/explicit.tex}
+\section{Comparing definitions of $A_\infty$ algebras}
+In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}.
+
+We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as:
+\begin{defn}
+A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with
+\begin{itemize}
+\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions
+\item and a compatible action of $\CD{[0,1]}$.
+\end{itemize}
+\end{defn}
+Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions.
+An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy).
+
+Translating between these definitions is straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels)
+$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$
+where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?}
+
+%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$.
+
+%\begin{defn}
+%A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'.
+
+%A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity
+%\begin{equation*}
+%f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}).
+%\end{equation*}
+
+%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that
+%\begin{enumerate}
+%\item The diagram
+%\begin{equation*}
+%\xymatrix{
+%\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\
+%\CD{[0,1]} \tensor A \ar[r]^{ev} & A
+%}
+%\end{equation*}
+%commutes up to weakly unique homotopy.
+%\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that
+%\begin{equation*}
+%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)).
+%\end{equation*}
+%\end{enumerate}
+%\end{defn}
+
+From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
+\begin{equation*}
+m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)).
+\end{equation*}
+
+It's then easy to calculate that
+\begin{align*}
+d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\
+ & = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\
+ & = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\
+\intertext{and thus that}
+m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1)
+\end{align*}
+as required (c.f. \cite[p. 6]{MR1854636}).
+\todo{then the general case.}
+We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.
+
+
+
\input{text/obsolete.tex}
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@@ -8,54 +8,64 @@
We begin with a definition of a \emph{topological $A_\infty$ category}, and then introduce the notion of a homological system of fields. A topological $A_\infty$ category gives a $1$-dimensional homological system of fields. We'll suggest that any good definition of a topological $A_\infty$ $n$-category with duals should allow construction of an $n$-dimensional homological system of fields, but we won't propose any such definition here. Later, we extend the definition of blob homology to allow homological fields as input. These definitions allow us to state and prove a theorem about the blob homology of a product manifold, and an intermediate theorem about gluing, in preparation for the proof of Property \ref{property:gluing}.
\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.
-First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$.
+\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
+\begin{itemize}
+\item for each pair of intervals $J_1$, $J_2$ so that $J_1 \cup_{\text{pt}} J_2$ is also an interval, `gluing' chain maps
+$$gl: \cC(J_1;a,b) \tensor \cC(J_2;b,c) \to \cC(J_1 \cup J_2;a,c),$$
+\item and `evaluation' chain maps $\CD{J \to J'} \tensor \cC(J;a,b) \to \cC(J';a,b)$
+\end{itemize}
+such that
+\begin{itemize}
+\item the gluing maps compose strictly associatively,
+\item the evaluation maps compose, up to a weakly unique homotopy,
+\item and the evaluation maps are compatible with the gluing maps, up to a weakly unique homotopy.
+\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 $J'$.
\begin{defn}
-A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'.
-
-A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity
-\begin{equation*}
-f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}).
-\end{equation*}
-
-An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that \todo{What goes here, if anything?}
-\begin{enumerate}
-\item The diagram
-\begin{equation*}
-\xymatrix{
-\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\
-\CD{[0,1]} \tensor A \ar[r]^{ev} & A
-}
-\end{equation*}
-commutes up to weakly unique \todo{???} homotopy.
-\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that
-\begin{equation*}
-\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)).
-\end{equation*}
-\end{enumerate}
+\label{defn:topological-Ainfty-module}%
+A \emph{topological $A_\infty$ left-module} $\cM$ over a topological $A_\infty$ category $\cC$ has for each right-marked interval $(J,p)$ and object $a \in \Obj(\cM)$ a chain complex $\cM(J,p; a)$, along with
+\begin{itemize}
+\item for each right-marked interval $(J,p)$, and interval $J'$ so that $J' \cup J$ is also right-marked interval, `gluing' chain maps
+$$gl: \cC(J';a,b) \tensor \cM(J,p;b) \to \cM(J' \cup J,p;a),$$
+\item and `evaluation' chain maps $\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \to \cM(J',p';a)$
+\end{itemize}
+satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}.
\end{defn}
-From a topological $A_\infty$ category $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define
-\begin{equation*}
-m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)).
-\end{equation*}
+A right module is the same, replacing right-marked intervals with left-marked intervals, and changing the order of the factors in the gluing maps.
-It's then easy to calculate that
-\begin{align*}
-d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\
- & = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\
- & = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\
-\intertext{and thus that}
-m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1)
-\end{align*}
-as required (c.f. \cite[p. 6]{MR1854636}).
-\todo{then the general case.}
-We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts.
+\begin{defn}
+\label{defn:topological-Ainfty-bimodule}%
+A \emph{topological $A_\infty$ bimodule} $\cM$ over a topological $A_\infty$ category $\cC$ has for each pair of a right-marked interval $(J,p)$ and a left-marked interval $(K,q)$ and object $a,b \in \Obj(\cM)$ a chain complex $\cM(J,p,K,q; a,b)$, along with
+\begin{itemize}
+\item for each pair of marked intervals $(J,p)$ and $(K,q)$, for each interval $J'$ so that $J' \cup J$ is also right-marked interval, a `gluing' chain maps
+$$gl: \cC(J';a',a) \tensor \cM(J,p,K,q;a,b) \to \cM(J' \cup J,p,K,q;a',b),$$
+and for each interval $K'$ so that $K \cup K'$ is also a left-marked interval, maps
+$$gl: \cM(J,p,K,q;a,b) \tensor \cC(K';b,b') \to \cM(J,p,K \cup K',q;a,b'),$$
+\item and `evaluation' chain maps $\CD{(J,p) \to (J',p')} \tensor \cM(J,p,K,q;a,b) \to \cM(J',p',K,q;a,b)$ and
+\end{itemize}
+satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}.
+\end{defn}
+
+
+\paragraph{Morphisms}
+
+\paragraph{Tensor products}
\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.
+
+
A topological $A_\infty$ category $\cC$ gives rise to a one dimensional homological system of fields. The functor $\cF_0$ simply assigns the set of objects of $\cC$ to a point.
For a $1$-manifold $X$, define a \emph{decomposition of $X$} with labels in $\cL$ as a (possibly empty) set of disjoint closed intervals $\{J\}$ in $X$, and a labeling of the complementary regions by elements of $\cL$.