--- 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}
-