--- a/text/A-infty.tex	Fri Oct 23 04:12:41 2009 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,198 +0,0 @@
-%!TEX root = ../blob1.tex
-
-\section{Homological systems of fields}
-\label{sec:homological-fields}
-
-\nn{*** If we keep Section \ref{sec:ncats}, then this section becomes obsolete.
-Retain it for now.}
-
-In this section, we extend the definition of blob homology to allow \emph{homological systems of fields}.
-
-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.
-
-\nn{And then we generalize all of this to $A_\infty$ $n$-categories [is this the
-best name for them?]}
-
-\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}
-
-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.)
-
-\nn{should say something about objects and restrictions of maps to boundaries of intervals 
-in next paragraph.}
-
-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}%
-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}
-
-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.
-
-\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}
-
-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.
-
-\nn{maybe should do a general interval instead of $[0,1]$.}
-
-\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}
-
-\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.
-
-
-
-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 $\cL$.
-
-The functor $\cF_1$ assigns to a $1$-manifold $X$ the vector space
-\begin{equation*}
-\cF_1(X) = \DirectSum_{\substack{\cJ \\ \text{a decomposition of $X$}}} \Tensor_{J \in \cJ} \cC_{l(J),r(J)}
-\end{equation*}
-where $l(J)$ and $r(J)$ denote the labels on the complementary regions on either side of the interval $J$. If $X$ has boundary, and we specify a boundary condition $c$ consisting of a label from $\Obj(\cC)$ at each boundary point, $\cF_1(X;c)$ is just the direct sum over decompositions agreeing with these boundary conditions. For any interval $I$, we define the local relations $\cU(I)$ to be the subcomplex of $\cF_1(I)$
-\begin{equation*}
-\cU(I) = \DirectSum_{\cJ} \ker\left(f_\cJ : \Tensor_{J \in \cJ} \cC_{l(J),r(J)} \to \cC_{l(I),r(I)} \right),
-\end{equation*}
-that is, the kernel of the composition map for $\cC$.
-
-\todo{explain why this satisfies the axioms}
-
-We now give two motivating examples, as theorems constructing other homological systems of fields,
-
-
-\begin{thm}
-For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as
-\begin{equation*}
-\Xi(M) = \CM{M}{X}.
-\end{equation*}
-\end{thm}
-
-\begin{thm}
-Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by
-\begin{equation*}
-\cF^{\times F}(M) = \cB_*(M \times F, \cF).
-\end{equation*}
-\end{thm}
-We might suggestively write $\cF^{\times F}$ as  $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories.
-
-
-In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields.
-
-
-\begin{thm}
-\begin{equation*}
-\cB_*(M, \Xi) \iso \Xi(M)
-\end{equation*}
-\end{thm}
-
-\begin{thm}[Product formula]
-Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields,
-there is a quasi-isomorphism
-\begin{align*}
-\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F})
-\end{align*}
-\end{thm}
-
-\begin{question}
-Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber?
-\end{question}
-
-\subsection{Blob homology}
-The definition of blob homology for $(\cF, \cU)$ a homological system of fields and local relations is essentially the same as that given before in \S \ref{???}, except now there are some extra terms in the differential accounting for the `internal' differential acting on the fields.
-
-As before
-\begin{equation*}
-	\cB_*^{\cF,\cU}(M) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
-		\left( \otimes_j \cU(B_j; c_j)\right) \otimes \cF(M \setmin B^t; c^t)
-\end{equation*}
-with $\overline{B}$ running over configurations of blobs satisfying the usual conditions, and $\overline{c}$ running over all boundary conditions. This is a doubly-graded vector space, graded by blob degree (the number of blobs) and internal degree (the sum of the homological degrees of the tensor factor fields). It becomes a complex by taking the homological degree to the be the sum of the blob and internal degrees, and defining $d$ by
-
-\begin{equation*}
-d f = \sum_{v \in t} \partial_v f + \sum_{v' \in t \cup \{\star\}} d_{v'} f,
-\end{equation*}
-
-
-%We'll write $\cT$ for the set of finite rooted trees. We'll think of each such a rooted tree as a category, with vertices as objects  and each morphism set either empty or a singleton, with $v \to w$ if $w$ is closer to a root of the tree than $v$. We'll write $\hat{v}$ for the `parent' of a vertex $v$ if $v$ is not a root (that is, $\hat{v}$ is the unique vertex such that $v \to \hat{v}$ but there is no $w$ with $v \to w \to \hat{v}$. If $v$ is a root, we'll write $\hat{v}=\star$. Further, for each tree $t$, let's arbitrarily choose an orientation $\lambda_t$, that is, an alternating $\pm1$-valued function on orderings of the vertices.
-
-%Given $v \in t$ there's a functor $\partial_v : t \to t \setminus \{v\}$ which removes the vertex $v$. Notice that removing a vertex naturally produces an orientation on $t \setminus \{v\}$ from the orientation on $t$, by $(\partial_v \lambda_t)(o) = \lambda_t(vo)$. This orientation may or may not agree with the chosen orientation of $t \setminus \{v\}$. We'll define $\sigma(v \in t) = \pm 1$ according to whether or not they agree. Notice that $$\sigma(v \in t) \sigma(w \in \partial_v t) = - \sigma(w \in t) \sigma(v \in \partial_w t).$$
-
-%Let $\operatorname{balls}(M)$ denote the category of open balls in $M$ with inclusions. Given a tree $t \in \cT$ we'll call a functor $b : t \to \operatorname{balls}(M)$ such that if $b(v) \cap b(v') \neq \emptyset$) then either $v \to v'$ or $v' \to v$, \emph{non-intersecting}.\footnote{Equivalently, if $b(v)$ and $b(v')$ are spanned in $\operatorname{balls}(M)$, then $v$ and $v'$ are spanned in $t$. That is, if there exists some ball $B \subset M$ so $B \subset b(v)$ and $B \subset b(v')$, then there must exist some $v'' \in t$ so $v'' \to v$ and $v'' \to v'$. Because $t$ is a tree, this implies either $v \to v'$ or $v' \to v$} For each non-intersecting functor $b$ define  
-%\begin{equation*}
-%\cF(t,b) = \cF\left(M \setminus b(t)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ not a leaf}}} \cF\left(b(v) \setminus b(v' \to v)\right)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ a leaf}}} \cU\left(b(v)\right)\right)
-%\end{equation*}
-%and then the vector space
-%\begin{equation*}
-%\cB_*^{\cF,\cU}(M) = \DirectSum_{t \in \cT} \DirectSum_{\substack{\text{non-intersecting}\\\text{functors} \\ b: t \to \operatorname{balls}(M)}} \cF(t,b)
-%\end{equation*}
-
-The blob degree of an element of $\cF(t,b)$ is the number of vertices in $t$, and the internal degree is the sum of the homological degrees in the tensor factors.
-The vector space $\cB_*^{\cF,\cU}(M)$ becomes a chain complex by taking the homological degree to be the sum of the blob and internal degrees, and defining $d$ on $\cF(t,b)$ by
-\begin{equation*}
-d f = \sum_{v \in t} \partial_v f + \sum_{v' \in t \cup \{\star\}} d_{v'} f,
-\end{equation*}
-where if $f \in \cF(t,b)$ is an elementary tensor of the form $f = f_\star \tensor \Tensor_{v \in t} f_v$ with
-\begin{align*}
-f_\star & \in \cF(M \setminus b(t)) && \\
-f_v       & \in \cF(b(v) \setminus b(v' \to v)) && \text{if $v$ is not a leaf} \\
-f_v       & \in \cU(b(v)) && \text{if $v$ is a leaf}
-\end{align*}
-the terms $\partial_v f$ are elementary tensors in $\cF(\partial_v t, \restrict{b}{\partial_v t})$ defined by
-\begin{equation*}
-(\partial_v f)_{v'} = \begin{cases} \sigma(v \in t) f_{\hat{v}} \circ f_v & \text{if $v' = \hat{v}$} \\ f_{v'} & \text{otherwise} \end{cases}
-\end{equation*}
-and the terms $d_v f$ are also elementary tensors in $\cF(t, b)$ defined by
-\begin{equation*}
-(d_v f)_{v'} = \begin{cases} (-1)^{\sum_{v \to v'} \deg f(v')} & \text{if $v'=v$} \\ f_v & \text{otherwise.} \end{cases}
-\end{equation*}
-
-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 $d^2=0$.}
-
-\input{text/smallblobs}
\ No newline at end of file

--- a/text/a_inf_blob.tex	Fri Oct 23 04:12:41 2009 +0000
+++ b/text/a_inf_blob.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -25,7 +25,9 @@
 new-fangled blob complex $\bc_*^\cF(Y)$.
 \end{thm}
 
-\begin{proof}
+\input{text/smallblobs}
+
+\begin{proof}[Proof of Theorem \ref{product_thm}]
 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
 
 First we define a map 
@@ -215,3 +217,30 @@
 \medskip
 \nn{still to do: fiber bundles, general maps}
 
+\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}
+
+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}
+

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/appendixes/comparing_defs.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -0,0 +1,193 @@
+%!TEX root = ../blob1.tex
+
+\section{Comparing $n$-category definitions}
+\label{sec:comparing-defs}
+
+In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats}
+to more traditional definitions, for $n=1$ and 2.
+
+\subsection{Plain 1-categories}
+
+Given a topological 1-category $\cC$, we construct a traditional 1-category $C$.
+(This is quite straightforward, but we include the details for the sake of completeness and
+to shed some light on the $n=2$ case.)
+
+Let the objects of $C$ be $C^0 \deq \cC(B^0)$ and the morphisms of $C$ be $C^1 \deq \cC(B^1)$, 
+where $B^k$ denotes the standard $k$-ball.
+The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$.
+
+Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
+Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree).
+By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule.
+Also by isotopy invariance, composition is associative.
+
+Given $a\in C^0$, define $\id_a \deq a\times B^1$.
+By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism.
+
+\nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?}
+
+\medskip
+
+For 1-categories based on oriented manifolds, there is no additional structure.
+
+For 1-categories based on unoriented manifolds, there is a map $*:C^1\to C^1$
+coming from $\cC$ applied to an orientation-reversing homeomorphism (unique up to isotopy) 
+from $B^1$ to itself.
+Topological properties of this homeomorphism imply that 
+$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$
+(* is an anti-automorphism).
+
+For 1-categories based on Spin manifolds,
+the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
+gives an order 2 automorphism of $C^1$.
+
+For 1-categories based on $\text{Pin}_-$ manifolds,
+we have an order 4 antiautomorphism of $C^1$.
+
+For 1-categories based on $\text{Pin}_+$ manifolds,
+we have an order 2 antiautomorphism and also an order 2 automorphism of $C^1$,
+and these two maps commute with each other.
+
+\nn{need to also consider automorphisms of $B^0$ / objects}
+
+\medskip
+
+In the other direction, given a traditional 1-category $C$
+(with objects $C^0$ and morphisms $C^1$) we will construct a topological
+1-category $\cC$.
+
+If $X$ is a 0-ball (point), let $\cC(X) \deq C^0$.
+If $S$ is a 0-sphere, let $\cC(S) \deq C^0\times C^0$.
+If $X$ is a 1-ball, let $\cC(X) \deq C^1$.
+Homeomorphisms isotopic to the identity act trivially.
+If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure
+to define the action of homeomorphisms not isotopic to the identity
+(and get, e.g., an unoriented topological 1-category).
+
+The domain and range maps of $C$ determine the boundary and restriction maps of $\cC$.
+
+Gluing maps for $\cC$ are determined my composition of morphisms in $C$.
+
+For $X$ a 0-ball, $D$ a 1-ball and $a\in \cC(X)$, define the product morphism 
+$a\times D \deq \id_a$.
+It is not hard to verify that this has the desired properties.
+
+\medskip
+
+The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back 
+more or less exactly the same thing we started with.  
+\nn{need better notation here}
+As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
+
+\medskip
+
+Similar arguments show that modules for topological 1-categories are essentially
+the same thing as traditional modules for traditional 1-categories.
+
+\subsection{Plain 2-categories}
+
+Let $\cC$ be a topological 2-category.
+We will construct a traditional pivotal 2-category.
+(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
+
+We will try to describe the construction in such a way the the generalization to $n>2$ is clear,
+though this will make the $n=2$ case a little more complicated than necessary.
+
+\nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons.
+Each approach has advantages and disadvantages.
+For better or worse, we choose bigons here.}
+
+\nn{maybe we should do both rectangles and bigons?}
+
+Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
+$k$-ball, which we also think of as the standard bihedron.
+Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
+into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
+Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
+whose boundary is splittable along $E$.
+This allows us to define the domain and range of morphisms of $C$ using
+boundary and restriction maps of $\cC$.
+
+Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.
+This is not associative, but we will see later that it is weakly associative.
+
+Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map 
+on $C^2$ (Figure \ref{fzo1}).
+Isotopy invariance implies that this is associative.
+We will define a ``horizontal" composition later.
+\nn{maybe no need to postpone?}
+
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.73}{tempkw/zo1}
+\end{equation*}
+\caption{Vertical composition of 2-morphisms}
+\label{fzo1}
+\end{figure}
+
+Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).
+Extended isotopy invariance for $\cC$ shows that this morphism is an identity for 
+vertical composition.
+
+Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.
+We will show that this 1-morphism is a weak identity.
+This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.
+Define let $a: y\to x$ be a 1-morphism.
+Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
+as shown in Figure \ref{fzo2}.
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.73}{tempkw/zo2}
+\end{equation*}
+\caption{blah blah}
+\label{fzo2}
+\end{figure}
+In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,
+while the remained is a half-pinched version of $a\times I$.
+\nn{the red region is unnecessary; remove it?  or does it help?
+(because it's what you get if you bigonify the natural rectangular picture)}
+We must show that the two compositions of these two maps give the identity 2-morphisms
+on $a$ and $a\bullet \id_x$, as defined above.
+Figure \ref{fzo3} shows one case.
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.83}{tempkw/zo3}
+\end{equation*}
+\caption{blah blah}
+\label{fzo3}
+\end{figure}
+In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}.
+\nn{also need to talk about (somewhere above) 
+how this sort of insertion is allowed by extended isotopy invariance and gluing.
+Also: maybe half-pinched and unpinched products can be derived from fully pinched
+products after all (?)}
+Figure \ref{fzo4} shows the other case.
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.83}{tempkw/zo4}
+\end{equation*}
+\caption{blah blah}
+\label{fzo4}
+\end{figure}
+We first collapse the red region, then remove a product morphism from the boundary,
+
+We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.
+It is not hard to show that this is independent of the arbitrary (left/right) choice made in the definition, and that it is associative.
+\begin{figure}[t]
+\begin{equation*}
+\mathfig{.83}{tempkw/zo5}
+\end{equation*}
+\caption{Horizontal composition of 2-morphisms}
+\label{fzo5}
+\end{figure}
+
+\nn{need to find a list of axioms for pivotal 2-cats to check}
+
+\nn{...}
+
+\medskip
+\hrule
+\medskip
+
+\nn{to be continued...}
+\medskip

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/appendixes/famodiff.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -0,0 +1,189 @@
+%!TEX root = ../blob1.tex
+
+\section{Families of Diffeomorphisms}  \label{sec:localising}
+
+Lo, the proof of Lemma (\ref{extension_lemma}):
+
+\nn{should this be an appendix instead?}
+
+\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in
+later draft}
+
+\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
+about boundary later}
+
+Recall that we are given
+an open cover $\cU = \{U_\alpha\}$ and an
+$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
+We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
+
+Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
+
+As a first approximation to the argument we will eventually make, let's replace $x$
+with a single singular cell
+\eq{
+    f: P \times X \to X .
+}
+Also, we'll ignore for now issues around $\bd P$.
+
+Our homotopy will have the form
+\eqar{
+    F: I \times P \times X &\to& X \\
+    (t, p, x) &\mapsto& f(u(t, p, x), x)
+}
+for some function
+\eq{
+    u : I \times P \times X \to P .
+}
+First we describe $u$, then we argue that it does what we want it to do.
+
+For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
+The various $K_\alpha$ should be in general position with respect to each other.
+We will see below that the $K_\alpha$'s need to be sufficiently fine in order
+to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not
+merely a homotopy through maps $X\to X$.
+
+Let $L$ be the union of all the $K_\alpha$'s.
+$L$ is itself a cell decomposition of $P$.
+\nn{next two sentences not needed?}
+To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$,
+where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$.
+Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$.
+
+Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
+Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
+more importantly, a $k{-}i$-dimensional normal coordinate.
+
+For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
+Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding
+$k$-cell of $L$.
+To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
+which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
+
+For $p \in D$ we define
+\eq{
+    u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
+}
+(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
+makes sense.)
+
+So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
+For handles of $J$ of index less than $k$, we will define $u$ to
+interpolate between the values on $k$-handles defined above.
+
+If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
+of $E$.
+In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
+with a $k$-handle.
+Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
+corresponding to $E$.
+Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
+adjacent to the $k{-}1$-cell corresponding to $E$.
+For $p \in E$, define
+\eq{
+    u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
+            + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
+}
+
+In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
+$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
+The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
+If we triangulate $R$ (without introducing new vertices), we can linearly extend
+a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
+Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
+the $k{-}j$-cell corresponding to $E$.
+For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
+Now define, for $p \in E$,
+\eq{
+    u(t, p, x) = (1-t)p + t \left(
+            \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
+                + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
+             \right) .
+}
+Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
+mentioned above.
+
+This completes the definition of $u: I \times P \times X \to P$.
+
+\medskip
+
+Next we verify that $u$ has the desired properties.
+
+Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
+Therefore $F$ is a homotopy from $f$ to something.
+
+Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,
+then $F$ is a homotopy through diffeomorphisms.
+We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
+We have
+\eq{
+%   \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
+    \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
+}
+Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
+\nn{bounded away from zero, or something like that}.
+(Recall that $X$ and $P$ are compact.)
+Also, $\pd{f}{p}$ is bounded.
+So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
+It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
+(which is bounded)
+and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
+These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
+This completes the proof that $F$ is a homotopy through diffeomorphisms.
+
+\medskip
+
+Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
+is a singular cell adapted to $\cU$.
+This will complete the proof of the lemma.
+\nn{except for boundary issues and the `$P$ is a cell' assumption}
+
+Let $j$ be the codimension of $D$.
+(Or rather, the codimension of its corresponding cell.  From now on we will not make a distinction
+between handle and corresponding cell.)
+Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
+where the $j_i$'s are the codimensions of the $K_\alpha$
+cells of codimension greater than 0 which intersect to form $D$.
+We will show that
+if the relevant $U_\alpha$'s are disjoint, then
+$F(1, \cdot, \cdot) : D\times X \to X$
+is a product of singular cells of dimensions $j_1, \ldots, j_m$.
+If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular
+cells whose dimensions correspond to a partition of the $j_i$'s.
+We will consider some simple special cases first, then do the general case.
+
+First consider the case $j=0$ (and $m=0$).
+A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,
+is independent of $p \in P$.
+So the corresponding map $D \to \Diff(X)$ is constant.
+
+Next consider the case $j = 1$ (and $m=1$, $j_1=1$).
+Now Equation yyyy applies.
+We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.
+It follows that the singular cell $D \to \Diff(X)$ can be written as a product
+of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
+The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
+
+Next case: $j=2$, $m=1$, $j_1 = 2$.
+This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
+1-dimensional.
+We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
+and a 2-cell with support $U_\beta$.
+
+Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.
+In this case the codimension 2 cell $D$ is the intersection of two
+codimension 1 cells, from $K_\beta$ and $K_\gamma$.
+We can write $D = D' \times I \times I$, where the normal coordinates are constant
+on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
+If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
+two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.
+If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and
+a 2-cell supported on $U_\beta \cup U_\gamma$.
+\nn{need to check that this is true}
+
+\nn{finally, general case...}
+
+\nn{this completes proof}
+
+\input{text/explicit.tex}
+

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/appendixes/misc_appendices.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -0,0 +1,91 @@
+%!TEX root = ../blob1.tex
+
+\section{Comparing definitions of $A_\infty$ algebras}
+\label{sec:comparing-A-infty}
+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.
+
+\section{Morphisms and duals of topological $A_\infty$ modules}
+\label{sec:A-infty-hom-and-duals}%
+
+\begin{defn}
+If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that  for each interval $J'$ the diagram
+\begin{equation*}
+\xymatrix{
+\cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\
+\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}}                                & \cN(J' cup J,a) 
+}
+\end{equation*}
+commutes on the nose, and the diagram
+\begin{equation*}
+\xymatrix{
+\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\
+\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}}  & \cN(J',p';a) \\
+}
+\end{equation*}
+commutes up to a weakly unique homotopy.
+\end{defn}
+
+The variations required for right modules and bimodules should be obvious.
+
+\todo{duals}
+\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic}
+

--- a/text/comparing_defs.tex	Fri Oct 23 04:12:41 2009 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,193 +0,0 @@
-%!TEX root = ../blob1.tex
-
-\section{Comparing $n$-category definitions}
-\label{sec:comparing-defs}
-
-In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats}
-to more traditional definitions, for $n=1$ and 2.
-
-\subsection{Plain 1-categories}
-
-Given a topological 1-category $\cC$, we construct a traditional 1-category $C$.
-(This is quite straightforward, but we include the details for the sake of completeness and
-to shed some light on the $n=2$ case.)
-
-Let the objects of $C$ be $C^0 \deq \cC(B^0)$ and the morphisms of $C$ be $C^1 \deq \cC(B^1)$, 
-where $B^k$ denotes the standard $k$-ball.
-The boundary and restriction maps of $\cC$ give domain and range maps from $C^1$ to $C^0$.
-
-Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
-Define composition in $C$ to be the induced map $C^1\times C^1 \to C^1$ (defined only when range and domain agree).
-By isotopy invariance in $\cC$, any other choice of homeomorphism gives the same composition rule.
-Also by isotopy invariance, composition is associative.
-
-Given $a\in C^0$, define $\id_a \deq a\times B^1$.
-By extended isotopy invariance in $\cC$, this has the expected properties of an identity morphism.
-
-\nn{(slash)id seems to rendering a a boldface 1 --- is this what we want?}
-
-\medskip
-
-For 1-categories based on oriented manifolds, there is no additional structure.
-
-For 1-categories based on unoriented manifolds, there is a map $*:C^1\to C^1$
-coming from $\cC$ applied to an orientation-reversing homeomorphism (unique up to isotopy) 
-from $B^1$ to itself.
-Topological properties of this homeomorphism imply that 
-$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$
-(* is an anti-automorphism).
-
-For 1-categories based on Spin manifolds,
-the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
-gives an order 2 automorphism of $C^1$.
-
-For 1-categories based on $\text{Pin}_-$ manifolds,
-we have an order 4 antiautomorphism of $C^1$.
-
-For 1-categories based on $\text{Pin}_+$ manifolds,
-we have an order 2 antiautomorphism and also an order 2 automorphism of $C^1$,
-and these two maps commute with each other.
-
-\nn{need to also consider automorphisms of $B^0$ / objects}
-
-\medskip
-
-In the other direction, given a traditional 1-category $C$
-(with objects $C^0$ and morphisms $C^1$) we will construct a topological
-1-category $\cC$.
-
-If $X$ is a 0-ball (point), let $\cC(X) \deq C^0$.
-If $S$ is a 0-sphere, let $\cC(S) \deq C^0\times C^0$.
-If $X$ is a 1-ball, let $\cC(X) \deq C^1$.
-Homeomorphisms isotopic to the identity act trivially.
-If $C$ has extra structure (e.g.\ it's a *-1-category), we use this structure
-to define the action of homeomorphisms not isotopic to the identity
-(and get, e.g., an unoriented topological 1-category).
-
-The domain and range maps of $C$ determine the boundary and restriction maps of $\cC$.
-
-Gluing maps for $\cC$ are determined my composition of morphisms in $C$.
-
-For $X$ a 0-ball, $D$ a 1-ball and $a\in \cC(X)$, define the product morphism 
-$a\times D \deq \id_a$.
-It is not hard to verify that this has the desired properties.
-
-\medskip
-
-The compositions of the above two ``arrows" ($\cC\to C\to \cC$ and $C\to \cC\to C$) give back 
-more or less exactly the same thing we started with.  
-\nn{need better notation here}
-As we will see below, for $n>1$ the compositions yield a weaker sort of equivalence.
-
-\medskip
-
-Similar arguments show that modules for topological 1-categories are essentially
-the same thing as traditional modules for traditional 1-categories.
-
-\subsection{Plain 2-categories}
-
-Let $\cC$ be a topological 2-category.
-We will construct a traditional pivotal 2-category.
-(The ``pivotal" corresponds to our assumption of strong duality for $\cC$.)
-
-We will try to describe the construction in such a way the the generalization to $n>2$ is clear,
-though this will make the $n=2$ case a little more complicated than necessary.
-
-\nn{Note: We have to decide whether our 2-morphsism are shaped like rectangles or bigons.
-Each approach has advantages and disadvantages.
-For better or worse, we choose bigons here.}
-
-\nn{maybe we should do both rectangles and bigons?}
-
-Define the $k$-morphisms $C^k$ of $C$ to be $\cC(B^k)_E$, where $B^k$ denotes the standard
-$k$-ball, which we also think of as the standard bihedron.
-Since we are thinking of $B^k$ as a bihedron, we have a standard decomposition of the $\bd B^k$
-into two copies of $B^{k-1}$ which intersect along the ``equator" $E \cong S^{k-2}$.
-Recall that the subscript in $\cC(B^k)_E$ means that we consider the subset of $\cC(B^k)$
-whose boundary is splittable along $E$.
-This allows us to define the domain and range of morphisms of $C$ using
-boundary and restriction maps of $\cC$.
-
-Choosing a homeomorphism $B^1\cup B^1 \to B^1$ defines a composition map on $C^1$.
-This is not associative, but we will see later that it is weakly associative.
-
-Choosing a homeomorphism $B^2\cup B^2 \to B^2$ defines a ``vertical" composition map 
-on $C^2$ (Figure \ref{fzo1}).
-Isotopy invariance implies that this is associative.
-We will define a ``horizontal" composition later.
-\nn{maybe no need to postpone?}
-
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.73}{tempkw/zo1}
-\end{equation*}
-\caption{Vertical composition of 2-morphisms}
-\label{fzo1}
-\end{figure}
-
-Given $a\in C^1$, define $\id_a = a\times I \in C^1$ (pinched boundary).
-Extended isotopy invariance for $\cC$ shows that this morphism is an identity for 
-vertical composition.
-
-Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.
-We will show that this 1-morphism is a weak identity.
-This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.
-Define let $a: y\to x$ be a 1-morphism.
-Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
-as shown in Figure \ref{fzo2}.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.73}{tempkw/zo2}
-\end{equation*}
-\caption{blah blah}
-\label{fzo2}
-\end{figure}
-In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,
-while the remained is a half-pinched version of $a\times I$.
-\nn{the red region is unnecessary; remove it?  or does it help?
-(because it's what you get if you bigonify the natural rectangular picture)}
-We must show that the two compositions of these two maps give the identity 2-morphisms
-on $a$ and $a\bullet \id_x$, as defined above.
-Figure \ref{fzo3} shows one case.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.83}{tempkw/zo3}
-\end{equation*}
-\caption{blah blah}
-\label{fzo3}
-\end{figure}
-In the first step we have inserted a copy of $id(id(x))$ \nn{need better notation for this}.
-\nn{also need to talk about (somewhere above) 
-how this sort of insertion is allowed by extended isotopy invariance and gluing.
-Also: maybe half-pinched and unpinched products can be derived from fully pinched
-products after all (?)}
-Figure \ref{fzo4} shows the other case.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.83}{tempkw/zo4}
-\end{equation*}
-\caption{blah blah}
-\label{fzo4}
-\end{figure}
-We first collapse the red region, then remove a product morphism from the boundary,
-
-We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}.
-It is not hard to show that this is independent of the arbitrary (left/right) choice made in the definition, and that it is associative.
-\begin{figure}[t]
-\begin{equation*}
-\mathfig{.83}{tempkw/zo5}
-\end{equation*}
-\caption{Horizontal composition of 2-morphisms}
-\label{fzo5}
-\end{figure}
-
-\nn{need to find a list of axioms for pivotal 2-cats to check}
-
-\nn{...}
-
-\medskip
-\hrule
-\medskip
-
-\nn{to be continued...}
-\medskip

--- a/text/evmap.tex	Fri Oct 23 04:12:41 2009 +0000
+++ b/text/evmap.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -1,6 +1,6 @@
 %!TEX root = ../blob1.tex
 
-\section{Action of $\CD{X}$}
+\section{Action of \texorpdfstring{$\CD{X}$}{$C_*(Diff(M))$}}
 \label{sec:evaluation}
 
 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of

--- a/text/famodiff.tex	Fri Oct 23 04:12:41 2009 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,189 +0,0 @@
-%!TEX root = ../blob1.tex
-
-\section{Families of Diffeomorphisms}  \label{sec:localising}
-
-Lo, the proof of Lemma (\ref{extension_lemma}):
-
-\nn{should this be an appendix instead?}
-
-\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in
-later draft}
-
-\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
-about boundary later}
-
-Recall that we are given
-an open cover $\cU = \{U_\alpha\}$ and an
-$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
-We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
-
-Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
-
-As a first approximation to the argument we will eventually make, let's replace $x$
-with a single singular cell
-\eq{
-    f: P \times X \to X .
-}
-Also, we'll ignore for now issues around $\bd P$.
-
-Our homotopy will have the form
-\eqar{
-    F: I \times P \times X &\to& X \\
-    (t, p, x) &\mapsto& f(u(t, p, x), x)
-}
-for some function
-\eq{
-    u : I \times P \times X \to P .
-}
-First we describe $u$, then we argue that it does what we want it to do.
-
-For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
-The various $K_\alpha$ should be in general position with respect to each other.
-We will see below that the $K_\alpha$'s need to be sufficiently fine in order
-to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not
-merely a homotopy through maps $X\to X$.
-
-Let $L$ be the union of all the $K_\alpha$'s.
-$L$ is itself a cell decomposition of $P$.
-\nn{next two sentences not needed?}
-To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$,
-where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$.
-Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$.
-
-Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
-Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
-more importantly, a $k{-}i$-dimensional normal coordinate.
-
-For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
-Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding
-$k$-cell of $L$.
-To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
-which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
-
-For $p \in D$ we define
-\eq{
-    u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
-}
-(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
-makes sense.)
-
-So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
-For handles of $J$ of index less than $k$, we will define $u$ to
-interpolate between the values on $k$-handles defined above.
-
-If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
-of $E$.
-In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
-with a $k$-handle.
-Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
-corresponding to $E$.
-Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
-adjacent to the $k{-}1$-cell corresponding to $E$.
-For $p \in E$, define
-\eq{
-    u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
-            + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
-}
-
-In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
-$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
-The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
-If we triangulate $R$ (without introducing new vertices), we can linearly extend
-a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
-Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
-the $k{-}j$-cell corresponding to $E$.
-For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
-Now define, for $p \in E$,
-\eq{
-    u(t, p, x) = (1-t)p + t \left(
-            \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
-                + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
-             \right) .
-}
-Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
-mentioned above.
-
-This completes the definition of $u: I \times P \times X \to P$.
-
-\medskip
-
-Next we verify that $u$ has the desired properties.
-
-Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
-Therefore $F$ is a homotopy from $f$ to something.
-
-Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,
-then $F$ is a homotopy through diffeomorphisms.
-We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
-We have
-\eq{
-%   \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
-    \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
-}
-Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
-\nn{bounded away from zero, or something like that}.
-(Recall that $X$ and $P$ are compact.)
-Also, $\pd{f}{p}$ is bounded.
-So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
-It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
-(which is bounded)
-and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
-These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
-This completes the proof that $F$ is a homotopy through diffeomorphisms.
-
-\medskip
-
-Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
-is a singular cell adapted to $\cU$.
-This will complete the proof of the lemma.
-\nn{except for boundary issues and the `$P$ is a cell' assumption}
-
-Let $j$ be the codimension of $D$.
-(Or rather, the codimension of its corresponding cell.  From now on we will not make a distinction
-between handle and corresponding cell.)
-Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
-where the $j_i$'s are the codimensions of the $K_\alpha$
-cells of codimension greater than 0 which intersect to form $D$.
-We will show that
-if the relevant $U_\alpha$'s are disjoint, then
-$F(1, \cdot, \cdot) : D\times X \to X$
-is a product of singular cells of dimensions $j_1, \ldots, j_m$.
-If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular
-cells whose dimensions correspond to a partition of the $j_i$'s.
-We will consider some simple special cases first, then do the general case.
-
-First consider the case $j=0$ (and $m=0$).
-A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,
-is independent of $p \in P$.
-So the corresponding map $D \to \Diff(X)$ is constant.
-
-Next consider the case $j = 1$ (and $m=1$, $j_1=1$).
-Now Equation yyyy applies.
-We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.
-It follows that the singular cell $D \to \Diff(X)$ can be written as a product
-of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
-The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
-
-Next case: $j=2$, $m=1$, $j_1 = 2$.
-This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
-1-dimensional.
-We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
-and a 2-cell with support $U_\beta$.
-
-Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.
-In this case the codimension 2 cell $D$ is the intersection of two
-codimension 1 cells, from $K_\beta$ and $K_\gamma$.
-We can write $D = D' \times I \times I$, where the normal coordinates are constant
-on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
-If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
-two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.
-If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and
-a 2-cell supported on $U_\beta \cup U_\gamma$.
-\nn{need to check that this is true}
-
-\nn{finally, general case...}
-
-\nn{this completes proof}
-
-\input{text/explicit.tex}
-

--- a/text/gluing.tex	Fri Oct 23 04:12:41 2009 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,324 +0,0 @@
-%!TEX root = ../blob1.tex
-
-\section{Gluing - needs to be rewritten/replaced}
-\label{sec:gluing}%
-
-\nn{*** this section is now obsolete; should be removed soon}
-
-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,
-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?}
-\nn{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.
-
-
-\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}
-

--- a/text/misc_appendices.tex	Fri Oct 23 04:12:41 2009 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,91 +0,0 @@
-%!TEX root = ../blob1.tex
-
-\section{Comparing definitions of $A_\infty$ algebras}
-\label{sec:comparing-A-infty}
-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.
-
-\section{Morphisms and duals of topological $A_\infty$ modules}
-\label{sec:A-infty-hom-and-duals}%
-
-\begin{defn}
-If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that  for each interval $J'$ the diagram
-\begin{equation*}
-\xymatrix{
-\cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\
-\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}}                                & \cN(J' cup J,a) 
-}
-\end{equation*}
-commutes on the nose, and the diagram
-\begin{equation*}
-\xymatrix{
-\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\
-\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}}  & \cN(J',p';a) \\
-}
-\end{equation*}
-commutes up to a weakly unique homotopy.
-\end{defn}
-
-The variations required for right modules and bimodules should be obvious.
-
-\todo{duals}
-\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic}
-

--- a/text/ncat.tex	Fri Oct 23 04:12:41 2009 +0000
+++ b/text/ncat.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -905,4 +905,55 @@
 \item morphisms of modules; show that it's adjoint to tensor product
 \end{itemize}
 
+\nn{Some salvaged paragraphs that we might want to work back in:}
+\hrule
 
+Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category 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 give two motivating examples, as theorems constructing other homological systems of fields,
+
+
+\begin{thm}
+For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as
+\begin{equation*}
+\Xi(M) = \CM{M}{X}.
+\end{equation*}
+\end{thm}
+
+\begin{thm}
+Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by
+\begin{equation*}
+\cF^{\times F}(M) = \cB_*(M \times F, \cF).
+\end{equation*}
+\end{thm}
+We might suggestively write $\cF^{\times F}$ as  $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories.
+
+
+In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields.
+
+
+\begin{thm}
+\begin{equation*}
+\cB_*(M, \Xi) \iso \Xi(M)
+\end{equation*}
+\end{thm}
+
+\begin{thm}[Product formula]
+Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields,
+there is a quasi-isomorphism
+\begin{align*}
+\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F})
+\end{align*}
+\end{thm}
+
+\begin{question}
+Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber?
+\end{question}
+
+\hrule

--- a/text/obsolete.tex	Fri Oct 23 04:12:41 2009 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,3 +0,0 @@
-\section{Obsolete stuff}
-
-\nn{nothing at the moment}

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/obsolete/A-infty.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -0,0 +1,196 @@
+%!TEX root = ../blob1.tex
+
+\section{Homological systems of fields}
+\label{sec:homological-fields}
+
+\nn{*** If we keep Section \ref{sec:ncats}, then this section becomes obsolete.
+Retain it for now.}
+
+In this section, we extend the definition of blob homology to allow \emph{homological systems of fields}.
+
+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.
+
+\nn{And then we generalize all of this to $A_\infty$ $n$-categories [is this the
+best name for them?]}
+
+\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}
+
+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.)
+
+\nn{should say something about objects and restrictions of maps to boundaries of intervals 
+in next paragraph.}
+
+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}%
+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}
+
+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.
+
+\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}
+
+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.
+
+\nn{maybe should do a general interval instead of $[0,1]$.}
+
+\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}
+
+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.
+
+
+
+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 $\cL$.
+
+The functor $\cF_1$ assigns to a $1$-manifold $X$ the vector space
+\begin{equation*}
+\cF_1(X) = \DirectSum_{\substack{\cJ \\ \text{a decomposition of $X$}}} \Tensor_{J \in \cJ} \cC_{l(J),r(J)}
+\end{equation*}
+where $l(J)$ and $r(J)$ denote the labels on the complementary regions on either side of the interval $J$. If $X$ has boundary, and we specify a boundary condition $c$ consisting of a label from $\Obj(\cC)$ at each boundary point, $\cF_1(X;c)$ is just the direct sum over decompositions agreeing with these boundary conditions. For any interval $I$, we define the local relations $\cU(I)$ to be the subcomplex of $\cF_1(I)$
+\begin{equation*}
+\cU(I) = \DirectSum_{\cJ} \ker\left(f_\cJ : \Tensor_{J \in \cJ} \cC_{l(J),r(J)} \to \cC_{l(I),r(I)} \right),
+\end{equation*}
+that is, the kernel of the composition map for $\cC$.
+
+\todo{explain why this satisfies the axioms}
+
+We now give two motivating examples, as theorems constructing other homological systems of fields,
+
+
+\begin{thm}
+For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as
+\begin{equation*}
+\Xi(M) = \CM{M}{X}.
+\end{equation*}
+\end{thm}
+
+\begin{thm}
+Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by
+\begin{equation*}
+\cF^{\times F}(M) = \cB_*(M \times F, \cF).
+\end{equation*}
+\end{thm}
+We might suggestively write $\cF^{\times F}$ as  $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories.
+
+
+In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields.
+
+
+\begin{thm}
+\begin{equation*}
+\cB_*(M, \Xi) \iso \Xi(M)
+\end{equation*}
+\end{thm}
+
+\begin{thm}[Product formula]
+Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields,
+there is a quasi-isomorphism
+\begin{align*}
+\cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F})
+\end{align*}
+\end{thm}
+
+\begin{question}
+Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber?
+\end{question}
+
+\subsection{Blob homology}
+The definition of blob homology for $(\cF, \cU)$ a homological system of fields and local relations is essentially the same as that given before in \S \ref{???}, except now there are some extra terms in the differential accounting for the `internal' differential acting on the fields.
+
+As before
+\begin{equation*}
+	\cB_*^{\cF,\cU}(M) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
+		\left( \otimes_j \cU(B_j; c_j)\right) \otimes \cF(M \setmin B^t; c^t)
+\end{equation*}
+with $\overline{B}$ running over configurations of blobs satisfying the usual conditions, and $\overline{c}$ running over all boundary conditions. This is a doubly-graded vector space, graded by blob degree (the number of blobs) and internal degree (the sum of the homological degrees of the tensor factor fields). It becomes a complex by taking the homological degree to the be the sum of the blob and internal degrees, and defining $d$ by
+
+\begin{equation*}
+d f = \sum_{v \in t} \partial_v f + \sum_{v' \in t \cup \{\star\}} d_{v'} f,
+\end{equation*}
+
+
+%We'll write $\cT$ for the set of finite rooted trees. We'll think of each such a rooted tree as a category, with vertices as objects  and each morphism set either empty or a singleton, with $v \to w$ if $w$ is closer to a root of the tree than $v$. We'll write $\hat{v}$ for the `parent' of a vertex $v$ if $v$ is not a root (that is, $\hat{v}$ is the unique vertex such that $v \to \hat{v}$ but there is no $w$ with $v \to w \to \hat{v}$. If $v$ is a root, we'll write $\hat{v}=\star$. Further, for each tree $t$, let's arbitrarily choose an orientation $\lambda_t$, that is, an alternating $\pm1$-valued function on orderings of the vertices.
+
+%Given $v \in t$ there's a functor $\partial_v : t \to t \setminus \{v\}$ which removes the vertex $v$. Notice that removing a vertex naturally produces an orientation on $t \setminus \{v\}$ from the orientation on $t$, by $(\partial_v \lambda_t)(o) = \lambda_t(vo)$. This orientation may or may not agree with the chosen orientation of $t \setminus \{v\}$. We'll define $\sigma(v \in t) = \pm 1$ according to whether or not they agree. Notice that $$\sigma(v \in t) \sigma(w \in \partial_v t) = - \sigma(w \in t) \sigma(v \in \partial_w t).$$
+
+%Let $\operatorname{balls}(M)$ denote the category of open balls in $M$ with inclusions. Given a tree $t \in \cT$ we'll call a functor $b : t \to \operatorname{balls}(M)$ such that if $b(v) \cap b(v') \neq \emptyset$) then either $v \to v'$ or $v' \to v$, \emph{non-intersecting}.\footnote{Equivalently, if $b(v)$ and $b(v')$ are spanned in $\operatorname{balls}(M)$, then $v$ and $v'$ are spanned in $t$. That is, if there exists some ball $B \subset M$ so $B \subset b(v)$ and $B \subset b(v')$, then there must exist some $v'' \in t$ so $v'' \to v$ and $v'' \to v'$. Because $t$ is a tree, this implies either $v \to v'$ or $v' \to v$} For each non-intersecting functor $b$ define  
+%\begin{equation*}
+%\cF(t,b) = \cF\left(M \setminus b(t)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ not a leaf}}} \cF\left(b(v) \setminus b(v' \to v)\right)\right) \tensor \left(\Tensor_{\substack{v \in t \\ \text{$v$ a leaf}}} \cU\left(b(v)\right)\right)
+%\end{equation*}
+%and then the vector space
+%\begin{equation*}
+%\cB_*^{\cF,\cU}(M) = \DirectSum_{t \in \cT} \DirectSum_{\substack{\text{non-intersecting}\\\text{functors} \\ b: t \to \operatorname{balls}(M)}} \cF(t,b)
+%\end{equation*}
+
+The blob degree of an element of $\cF(t,b)$ is the number of vertices in $t$, and the internal degree is the sum of the homological degrees in the tensor factors.
+The vector space $\cB_*^{\cF,\cU}(M)$ becomes a chain complex by taking the homological degree to be the sum of the blob and internal degrees, and defining $d$ on $\cF(t,b)$ by
+\begin{equation*}
+d f = \sum_{v \in t} \partial_v f + \sum_{v' \in t \cup \{\star\}} d_{v'} f,
+\end{equation*}
+where if $f \in \cF(t,b)$ is an elementary tensor of the form $f = f_\star \tensor \Tensor_{v \in t} f_v$ with
+\begin{align*}
+f_\star & \in \cF(M \setminus b(t)) && \\
+f_v       & \in \cF(b(v) \setminus b(v' \to v)) && \text{if $v$ is not a leaf} \\
+f_v       & \in \cU(b(v)) && \text{if $v$ is a leaf}
+\end{align*}
+the terms $\partial_v f$ are elementary tensors in $\cF(\partial_v t, \restrict{b}{\partial_v t})$ defined by
+\begin{equation*}
+(\partial_v f)_{v'} = \begin{cases} \sigma(v \in t) f_{\hat{v}} \circ f_v & \text{if $v' = \hat{v}$} \\ f_{v'} & \text{otherwise} \end{cases}
+\end{equation*}
+and the terms $d_v f$ are also elementary tensors in $\cF(t, b)$ defined by
+\begin{equation*}
+(d_v f)_{v'} = \begin{cases} (-1)^{\sum_{v \to v'} \deg f(v')} & \text{if $v'=v$} \\ f_v & \text{otherwise.} \end{cases}
+\end{equation*}
+
+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 $d^2=0$.}
+

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/obsolete/gluing.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -0,0 +1,324 @@
+%!TEX root = ../blob1.tex
+
+\section{Gluing - needs to be rewritten/replaced}
+\label{sec:gluing}%
+
+\nn{*** this section is now obsolete; should be removed soon}
+
+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,
+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?}
+\nn{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.
+
+
+\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}
+

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/obsolete/obsolete.tex	Mon Oct 26 05:39:29 2009 +0000
@@ -0,0 +1,15 @@
+\documentclass[11pt,leqno]{amsart}
+
+%\usepackage{amsthm}
+
+\newcommand{\pathtotrunk}{../../}
+\input{\pathtotrunk text/article_preamble}
+\input{\pathtotrunk text/top_matter}
+\input{\pathtotrunk text/kw_macros}
+
+\begin{document}
+
+ \input{A-infty}
+ \input{gluing}
+
+\end{document}