--- a/text/a_inf_blob.tex	Wed Oct 28 00:54:35 2009 +0000
+++ b/text/a_inf_blob.tex	Wed Oct 28 02:44:29 2009 +0000
@@ -2,6 +2,7 @@
 
 \section{The blob complex for $A_\infty$ $n$-categories}
 \label{sec:ainfblob}
+\label{sec:gluing}
 
 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob
 complex $\bc_*(M)$ to the be the colimit $\cC(M)$ of Section \ref{sec:ncats}.

--- a/text/basic_properties.tex	Wed Oct 28 00:54:35 2009 +0000
+++ b/text/basic_properties.tex	Wed Oct 28 02:44:29 2009 +0000
@@ -27,7 +27,7 @@
 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
 of the quotient map
 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
-For example, this is always the case if you coefficient ring is a field.
+For example, this is always the case if the coefficient ring is a field.
 Then
 \begin{prop} \label{bcontract}
 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$
@@ -66,14 +66,14 @@
 
 \begin{prop}
 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
-of $n$-manifolds and diffeomorphisms to the category of chain complexes and
+of $n$-manifolds and homeomorphisms to the category of chain complexes and
 (chain map) isomorphisms.
 \qed
 \end{prop}
 
 In particular,
 \begin{prop}  \label{diff0prop}
-There is an action of $\Diff(X)$ on $\bc_*(X)$.
+There is an action of $\Homeo(X)$ on $\bc_*(X)$.
 \qed
 \end{prop}
 
@@ -106,16 +106,16 @@
 The above map is very far from being an isomorphism, even on homology.
 This will be fixed in Section \ref{sec:gluing} below.
 
-\nn{Next para not need, since we already use bullet = gluing notation above(?)}
+%\nn{Next para not needed, since we already use bullet = gluing notation above(?)}
 
-An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
-and $X\sgl = X_1 \cup_Y X_2$.
-(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
-For $x_i \in \bc_*(X_i)$, we introduce the notation
-\eq{
-    x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
-}
-Note that we have resumed our habit of omitting boundary labels from the notation.
+%An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
+%and $X\sgl = X_1 \cup_Y X_2$.
+%(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
+%For $x_i \in \bc_*(X_i)$, we introduce the notation
+%\eq{
+%    x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
+%}
+%Note that we have resumed our habit of omitting boundary labels from the notation.
 
 
 

--- a/text/definitions.tex	Wed Oct 28 00:54:35 2009 +0000
+++ b/text/definitions.tex	Wed Oct 28 02:44:29 2009 +0000
@@ -105,10 +105,21 @@
 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.
 \end{enumerate}
 
-\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$}
+There are two notations we commonly use for gluing.
+One is 
+\[
+	x\sgl \deq \gl(x) \in \cC(X\sgl) , 
+\]
+for $x\in\cC(X)$.
+The other is
+\[
+	x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , 
+\]
+in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$.
 
-\bigskip
-Using the functoriality and $\bullet\times I$ properties above, together
+\medskip
+
+Using the functoriality and $\cdot\times I$ properties above, together
 with boundary collar homeomorphisms of manifolds, we can define the notion of 
 {\it extended isotopy}.
 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold

--- a/text/evmap.tex	Wed Oct 28 00:54:35 2009 +0000
+++ b/text/evmap.tex	Wed Oct 28 02:44:29 2009 +0000
@@ -5,7 +5,7 @@
 
 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of
 the space of diffeomorphisms
-\nn{or homeomorphisms}
+\nn{or homeomorphisms; need to fix the diff vs homeo inconsistency}
 between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$).
 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general
 than simplices --- they can be based on any linear polyhedron.
@@ -19,14 +19,22 @@
 }
 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$
 (Proposition (\ref{diff0prop})).
-For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, 
+For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, 
 the following diagram commutes up to homotopy
 \eq{ \xymatrix{
-     CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}}    & \bc_*(Y) \\
-     CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
-        \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}}  \ar[u]^{\gl \otimes \gl}  &
-            \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
+     CD_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}}    & \bc_*(Y\sgl) \\
+      CD_*(X, Y) \otimes \bc_*(X)
+        \ar@/_4ex/[r]_{e_{XY}}  \ar[u]^{\gl \otimes \gl}  &
+            \bc_*(Y) \ar[u]_{\gl}
 } }
+%For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, 
+%the following diagram commutes up to homotopy
+%\eq{ \xymatrix{
+%     CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}}    & \bc_*(Y) \\
+%     CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+%        \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}}  \ar[u]^{\gl \otimes \gl}  &
+%            \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
+%} }
 Any other map satisfying the above two properties is homotopic to $e_X$.
 \end{prop}
 

--- a/text/hochschild.tex	Wed Oct 28 00:54:35 2009 +0000
+++ b/text/hochschild.tex	Wed Oct 28 02:44:29 2009 +0000
@@ -3,11 +3,16 @@
 \section{Hochschild homology when $n=1$}
 \label{sec:hochschild}
 
+So far we have provided no evidence that blob homology is interesting in degrees 
+greater than zero.
 In this section we analyze the blob complex in dimension $n=1$
 and find that for $S^1$ the blob complex is homotopy equivalent to the 
 Hochschild complex of the category (algebroid) that we started with.
+Thus the blob complex is a natural generalization of something already
+known to be interesting in higher homological degrees.
 
-\nn{initial idea for blob complex came from thinking about...}
+It is also worth noting that the original idea for the blob complex came from trying
+to find a more ``local" description of the Hochschild complex.
 
 \nn{need to be consistent about quasi-isomorphic versus homotopy equivalent
 in this section.
@@ -38,7 +43,8 @@
 Hochschild complex of $C$.
 Note that both complexes are free (and hence projective), so it suffices to show that they
 are quasi-isomorphic.
-In order to prove this we will need to extend the blob complex to allow points to also
+In order to prove this we will need to extend the 
+definition of the blob complex to allow points to also
 be labeled by elements of $C$-$C$-bimodules.
 
 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
@@ -223,6 +229,7 @@
 Suppose we have a short exact sequence of $C$-$C$ bimodules $$\xymatrix{0 \ar[r] & K \ar@{^{(}->}[r]^f & E \ar@{->>}[r]^g & Q \ar[r] & 0}.$$
 We'll write $\hat{f}$ and $\hat{g}$ for the image of $f$ and $g$ under the functor.
 Most of what we need to check is easy.
+\nn{don't we need to consider sums here, e.g. $\sum_i(a_i\ot k_i\ot b_i)$ ?}
 If $(a \tensor k \tensor b) \in \ker(C \tensor K \tensor C \to K)$, to have $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$, we must have $f(k) = 0 \in E$, which implies $k=0$ itself. If $(a \tensor e \tensor b) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, clearly
 $e$ is in the image of the original $f$, so is in the kernel of the original $g$, and so $\hat{g}(a \tensor e \tensor b) = 0$.
 If $\hat{g}(a \tensor e \tensor b) = 0$, then $g(e) = 0$, so $e = f(\widetilde{e})$ for some $\widetilde{e} \in K$, and $a \tensor e \tensor b = \hat{f}(a \tensor \widetilde{e} \tensor b)$.

--- a/text/ncat.tex	Wed Oct 28 00:54:35 2009 +0000
+++ b/text/ncat.tex	Wed Oct 28 02:44:29 2009 +0000
@@ -2,14 +2,17 @@
 
 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
 
-\section{$n$-categories (maybe)}
+\section{$n$-categories}
 \label{sec:ncats}
 
-\nn{experimental section.  maybe this should be rolled into other sections.
-maybe it should be split off into a separate paper.}
+%In order to make further progress establishing properties of the blob complex,
+%we need a definition of $A_\infty$ $n$-category that is adapted to our needs.
+%(Even in the case $n=1$, we need the new definition given below.)
+%It turns out that the $A_\infty$ $n$-category definition and the plain $n$-category
+%definition are mostly the same, so we give a new definition of plain
+%$n$-categories too.
+%We also define modules and tensor products for both plain and $A_\infty$ $n$-categories.
 
-\nn{comment somewhere that what we really need is a convenient def of infty case, including tensor products etc.
-but while we're at it might as well do plain case too.}
 
 \subsection{Definition of $n$-categories}
 
@@ -18,6 +21,16 @@
 (As is the case throughout this paper, by ``$n$-category" we mean
 a weak $n$-category with strong duality.)
 
+The definitions presented below tie the categories more closely to the topology
+and avoid combinatorial questions about, for example, the minimal sufficient
+collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets.
+For examples of topological origin, it is typically easy to show that they
+satisfy our axioms.
+For examples of a more purely algebraic origin, one would typically need the combinatorial
+results that we have avoided here.
+
+\medskip
+
 Consider first ordinary $n$-categories.
 We need a set (or sets) of $k$-morphisms for each $0\le k \le n$.
 We must decide on the ``shape" of the $k$-morphisms.
@@ -52,6 +65,7 @@
 So we replace the above with
 
 \xxpar{Morphisms:}
+%\xxpar{Axiom 1 -- Morphisms:}
 {For each $0 \le k \le n$, we have a functor $\cC_k$ from 
 the category of $k$-balls and 
 homeomorphisms to the category of sets and bijections.}
@@ -116,6 +130,7 @@
 equipped with an orientation of its once-stabilized tangent bundle.
 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of 
 their $k$ times stabilized tangent bundles.
+(cf. [Stolz and Teichner].)
 Probably should also have a framing of the stabilized dimensions in order to indicate which 
 side the bounded manifold is on.
 For the moment just stick with unoriented manifolds.}