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+++ b/text/gluing.tex Tue Feb 03 21:06:04 2009 +0000
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+\section{Gluing - needs to be rewritten/replaced}
+\label{sec:gluing}%
+
+We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction
+\begin{itemize}
+%\mbox{}% <-- gets the indenting right
+\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is
+naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
+
+\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an
+$A_\infty$ module for $\bc_*(Y \times I)$.
+
+\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension
+$0$-submanifold of its boundary, the blob homology of $X'$, obtained from
+$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of
+$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.
+\begin{equation*}
+\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
+\end{equation*}
+\end{itemize}
+
+Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative
+definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions,
+and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes
+the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras
+as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the
+Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the
+self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs
+of the gluing formula in the topological context.
+
+\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
+
+\begin{thm}
+Topological $A_\infty$-$1$-categories are equivalent to the usual notion of
+$A_\infty$-$1$-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,\scott{Why are we treating the $n>1$ cases at all?} 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
+}
+
+\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
+$I=\left[0,1\right]$, 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
+$\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$.
+\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}
+This data is required to satisfy the following conditions.
+\begin{itemize}
+\item The evaluation chain map is associative, in that the diagram
+\begin{equation*}
+\xymatrix{
+ & \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\
+\CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\
+ & A(J'') &
+}
+\end{equation*}
+commutes 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:
+%\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.
+\item The gluing and evaluation maps are compatible.
+\nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?}
+\end{itemize}
+\end{defn}
+
+\begin{rem}
+We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together
+constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces.
+Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$.
+\end{rem}
+
+%% if we do things separately, we should say this:
+%\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 of $k$-parameter families of diffeomorphisms) ignores the boundary conditions.
+\todo{we presumably need to say something about $\id_c \in A(J, c, c)$.}
+
+At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$.
+\begin{defn}
+Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by
+\begin{enumerate}
+\item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$,
+\item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ 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,
+\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.
+\end{defn}
+
+The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$.
+Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties
+\ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra.
+
+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 following data.
+\begin{enumerate}
+\item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces.
+\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
+conditions analogous to those in Definition \ref{defn:topological-algebra}.
+\end{defn}
+
+For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as
+a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above.
+For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$.
+(Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties
+\ref{property:evaluation} and \ref{property:gluing-map} respectively.
+
+The definition of a bimodule is like the definition of a module,
+except that we have two disjoint marked intervals $K$ and $L$, one with a marked point
+on the upper boundary and the other with a marked point on the lower boundary.
+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
+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
+structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$.
+
+Next we define the coend
+(or gluing or tensor product or self tensor product, depending on the context)
+$\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties.
+\begin{defn}
+We define a category $\cG(M)$. Objects consist of the following data.
+\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
+$\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$.
+\item For each decomposition of intervals $N = K\cup L$,
+a gluing map $\gl_{K,L} : M(K,L) \to C(N)$.
+\end{itemize}
+This data must satisfy the following conditions.
+\begin{itemize}
+\item The evaluation maps are associative.
+\nn{up to homotopy?}
+\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.
+\item the gluing and evaluation maps are compatible.
+\end{itemize}
+
+A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked,
+satisfying the following conditions.
+\begin{itemize}
+\item For each pair of intervals $N,N'$, the diagram
+\begin{equation*}
+\xymatrix{
+\CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\
+C(N) \ar[r]_{f_N} & C'(N)
+}
+\end{equation*}
+commutes.
+\item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition
+$$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$
+\end{itemize}
+\end{defn}
+
+We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$,
+there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$
+factor through the gluing maps for $\gl(M)$.
+
+We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself
+is essentially $C_*(\Maps(S^1 \to M))$. \todo{}
+
+For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment
+$$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$.
+Showing that it is an initial object is the content of the gluing theorem proved below.
+
+The definitions for a topological $A_\infty$-$n$-category are very similar to the above
+$n=1$ case.
+One replaces intervals with manifolds diffeomorphic to the ball $B^n$.
+Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$.
+
+\nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.}
+
+\todo{the motivating example $C_*(\maps(X, M))$}
+
+
+
+\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 as a 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.
+
+We'll first define $\cT_{k,n}$ to be the set of planar forests consisting of $n-k$ trees, with a total of $n$ leaves. Thus
+\todo{$\cT_{0,n}$ has 1 element, with $n$ vertical lines, $\cT_{1,n}$ has $n-1$ elements, each with a single trivalent vertex, $\cT_{2,n}$ etc...}
+\begin{align*}
+\end{align*}
+
+\begin{defn}
+The topological $A_\infty$ category $\bc_*^A$ is doubly graded, by `blob degree' and `internal degree'. We'll write $\bc_k^A$ for the blob degree $k$ piece.
+The homological degree of an element $a \in \bc_*^A(J)$
+is the sum of the blob degree and the internal degree.
+
+We first define $\bc_0^A(J)$ as a vector space by
+\begin{equation*}
+\bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A).
+\end{equation*}
+(That is, for each division of $J$ into finitely many subintervals,
+we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval,
+and a copy of $A$ for each subinterval.)
+The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains
+plus the sum of the homological degrees of the elements of $A$.
+The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$.
+
+Next,
+\begin{equation*}
+\bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A).
+\end{equation*}
+\end{defn}
+
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{0.7}{associahedron/A4-vertices}
+\end{equation*}
+\caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.}
+\label{fig:A4-vertices}
+\end{figure}
+
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{0.7}{associahedron/A4-faces}
+\end{equation*}
+\caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.}
+\label{fig:A4-vertices}
+\end{figure}
+
+\newcommand{\tm}{\widetilde{m}}
+
+Let $\tm_1(a) = a$.
+
+We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it.
+\begin{align}
+\notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\
+\label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\
+\label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\
+\label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k)
+\end{align}
+The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$.
+The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves.
+Note here that we have one more leaf than there arguments of $\tm_k$.
+(See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.)
+We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves.
+The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both
+vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form
+$$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$
+where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices.
+If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$
+in \eqref{eq:bdy-tm-k-3},
+where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex.
+For example, we have
+\begin{align*}
+\bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\
+ & \qquad - a \tensor b + m_2(a \tensor b) \\
+\bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\
+ & \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\
+ & \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right)
+\end{align*}
+\begin{align*}
+\bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\
+ & + \left(\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)\right) + \\
+ & + \left(\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d))\right. + \\
+ & + \left.\tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)\right) \\
+\end{align*}
+See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest
+to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one.
+
+\begin{figure}[!ht]
+\begin{equation*}
+\mathfig{1.0}{associahedron/A4-terms}
+\end{equation*}
+\caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.}
+\label{fig:A4-terms}
+\end{figure}
+
+\begin{lem}
+This definition actually results in a chain complex, that is $\bdy^2 = 0$.
+\end{lem}
+\begin{proof}
+\newcommand{\T}{\text{---}}
+\newcommand{\ssum}[1]{{\sum}^{(#1)}}
+For the duration of this proof, inside a summation over variables $l_1, \ldots, l_m$, an expression with $m$ dashes will be interpreted
+by replacing each dash with contiguous factors from $a_1 \tensor \cdots \tensor a_k$, so the first dash takes the first $l_1$ factors, the second
+takes the next $l_2$ factors, and so on. Further, we'll write $\ssum{m}$ for $\sum_{\sum_{i=1}^m l_i = k}$.
+In this notation, the formula for the differential becomes
+\begin{align}
+\notag
+\bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\
+\intertext{and we calculate}
+\notag
+\bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\
+\notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\
+\notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\
+\label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\
+\label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\
+\label{eq:d23} & \qquad + \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2+l_3} \sigma_{l_1;l_2,l_3} \\
+\label{eq:d24} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \sigma_{0;l_1,l_2+l_3+l_4} \tau_{l_1;l_2,l_3,l_4} \\
+\label{eq:d25} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \tau_{0;l_1,l_2,l_3+l_4} ??? \\
+\label{eq:d26} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2,l_3,l_4} \sigma_{0;l_1,l_2} \\
+\label{eq:d27} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2+l_3,l_4,l_5} \tau_{0;l_1,l_2,l_3} \\
+\label{eq:d28} & \qquad + \ssum{5} \tm(\T \tensor m(\T \tensor m(\T) \tensor \T) \tensor \T) \times \tau_{0;l_1,l_2+l_3+l_4,l_5} ??? \\
+\label{eq:d29} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3+l_4+l_5} ???
+\end{align}
+Now, we see the the expressions on the right hand side of line \eqref{eq:d21} and those on \eqref{eq:d23} cancel. Similarly, line \eqref{eq:d22} cancels
+with \eqref{eq:d25}, \eqref{eq:d24} with \eqref{eq:d26}, and \eqref{eq:d27} with \eqref{eq:d29}. Finally, we need to see that \eqref{eq:d28} gives $0$,
+by the usual relations between the $m_k$ in an $A_\infty$ algebra.
+\end{proof}
+
+\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
+easy, I think, so maybe it should be done earlier??}
+
+\bigskip
+
+Outline:
+\begin{itemize}
+\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product.
+use graphical/tree point of view, rather than following Keller exactly
+\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration?
+\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing
+\item motivating example: $C_*(\maps(X, M))$
+\item maybe incorporate dual point of view (for $n=1$), where points get
+object labels and intervals get 1-morphism labels
+\end{itemize}
+
+
+\subsection{$A_\infty$ action on the boundary}
+\label{sec:boundary-action}%
+Let $Y$ be an $n{-}1$-manifold.
+The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
+conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure
+of an $A_\infty$ category.
+
+Composition of morphisms (multiplication) depends of a choice of homeomorphism
+$I\cup I \cong I$. Given this choice, gluing gives a map
+\eq{
+ \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
+ \cong \bc_*(Y\times I; a, c)
+}
+Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various
+higher associators of the $A_\infty$ structure, more or less canonically.
+
+\nn{is this obvious? does more need to be said?}
+
+Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$.
+
+Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism
+$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$
+(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
+$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$.
+Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood
+of $Y$ in $X$.
+
+In the next section we use the above $A_\infty$ actions to state and prove
+a gluing theorem for the blob complexes of $n$-manifolds.
+
+
+\subsection{The gluing formula}
+\label{sec:gluing-formula}%
+Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy
+of $Y \du -Y$ contained in its boundary.
+Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$.
+We wish to describe the blob complex of $X\sgl$ in terms of the blob complex
+of $X$.
+More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
+where $c\sgl \in \cC(\bd X\sgl)$,
+in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation
+of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$.
+
+\begin{thm}
+$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product
+of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
+\end{thm}
+
+The proof will occupy the remainder of this section.
+
+\nn{...}
+
+\bigskip
+
+\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category}
+