--- a/preamble.tex	Tue Dec 08 01:08:53 2009 +0000
+++ b/preamble.tex	Fri Dec 11 22:44:25 2009 +0000
@@ -70,6 +70,8 @@
 \newtheorem*{defn*}{Definition}             % unnumbered definition
 \newtheorem{question}{Question}
 \newtheorem{property}{Property}
+\newtheorem{axiom}{Axiom}
+\newenvironment{preliminary-axiom}[2]{\textbf{Axiom #1 [preliminary] (#2)}\it}{}
 \newenvironment{rem}{\noindent\textsl{Remark.}}{}  % perhaps looks better than rem above?
 \numberwithin{equation}{section}
 %\numberwithin{figure}{section}

--- a/text/intro.tex	Tue Dec 08 01:08:53 2009 +0000
+++ b/text/intro.tex	Fri Dec 11 22:44:25 2009 +0000
@@ -24,17 +24,16 @@
 
 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category.
 
-Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It's not clear that we could remove the duality conditions from our definition, even if we wanted to.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
+Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
 
-\nn{Not sure that the next para is appropriate here}
-The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible; a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. This vector spaces glue together associatively. For an $A_\infty$ $n$-category, we instead associate a chain complex to each such $B$. We require that homeomorphisms (or the complex of singular chains of homeomorphisms in the $A_\infty$ case) act. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls, using a topological $n$-category, and the complex $\CM{-}{X}$ of maps to a fixed target space $X$.
-\nn{we might want to make our choice of notation here ($B$, $X$) consistent with later sections ($X$, $T$), or vice versa}
+The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism group.
+For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
 
-In  \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $n$-manifold and an $A_\infty$ $n$-category. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
+In  \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below.
 
 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.}
 
-Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
+Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
 
 
 \nn{some more things to cover in the intro}
@@ -123,7 +122,7 @@
 \begin{property}[Functoriality]
 \label{property:functoriality}%
 The blob complex is functorial with respect to homeomorphisms. That is, 
-for fixed $n$-category / fields $\cC$, the association
+for a fixed $n$-dimensional system of fields $\cC$, the association
 \begin{equation*}
 X \mapsto \bc_*^{\cC}(X)
 \end{equation*}
@@ -150,9 +149,9 @@
 Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is
 a natural map
 \[
-	\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) .
+	\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) 
 \]
-(Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.)
+(natural with respect to homeomorphisms, and also associative with respect to iterated gluings).
 \end{property}
 
 \begin{property}[Contractibility]
@@ -161,7 +160,7 @@
 the map to 0-th homology has a splitting; need to fix statement}
 The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$.
 \begin{equation}
-\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))}
+\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
 \end{equation}
 \end{property}
 
@@ -180,7 +179,7 @@
 The blob complex for a $1$-category $\cC$ on the circle is
 quasi-isomorphic to the Hochschild complex.
 \begin{equation*}
-\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)}
+\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
 \end{equation*}
 \end{property}
 
@@ -204,7 +203,6 @@
 }
 \end{equation*}
 \nn{should probably say something about associativity here (or not?)}
-\nn{maybe do self-gluing instead of 2 pieces case:}
 Further, for
 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram
 (using the gluing maps described in Property \ref{property:gluing-map}) commutes.
@@ -220,11 +218,10 @@
 
 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category.
 
-\begin{property}[Blob complexes of balls form an $A_\infty$ $n$-category]
+\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category]
 \label{property:blobs-ainfty}
 Let $\cC$ be  a topological $n$-category.  Let $Y$ be an $n{-}k$-manifold. 
-Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m \leq k$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category.
-\nn{the subscript * is only appropriate when $m=k$. }
+Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category.
 \end{property}
 \begin{rem}
 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category.
@@ -235,14 +232,14 @@
 
 \begin{property}[Product formula]
 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category.
-Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
+Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}).
 Then
 \[
-	\bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y)) .
+	\bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) .
 \]
 Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories.
 \end{property}
-It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework as such a statement.
+It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework for such a statement.
 
 \begin{property}[Gluing formula]
 \label{property:gluing}%
@@ -262,22 +259,23 @@
 \end{itemize}
 \end{property}
 
-
+Finally, we state two more properties, which we will not prove in this paper.
 
 \begin{property}[Mapping spaces]
-Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps 
-$B^n \to W$.
-(The case $n=1$ is the usual $A_\infty$ category of paths in $W$.)
+Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps 
+$B^n \to T$.
+(The case $n=1$ is the usual $A_\infty$ category of paths in $T$.)
 Then 
-$$\bc_*(M, \pi^\infty_{\le n}(W) \simeq \CM{M}{W}.$$
+$$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$
 \end{property}
 
+This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data.
+
 \begin{property}[Higher dimensional Deligne conjecture]
 \label{property:deligne}
 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
 \end{property}
-See \S \ref{sec:deligne} for an explanation of the terms appearing here, and (in a later edition of this paper) the proof.
-
+See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere.
 
 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in
 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.}

--- a/text/ncat.tex	Tue Dec 08 01:08:53 2009 +0000
+++ b/text/ncat.tex	Fri Dec 11 22:44:25 2009 +0000
@@ -18,8 +18,8 @@
 
 Before proceeding, we need more appropriate definitions of $n$-categories, 
 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
-(As is the case throughout this paper, by ``$n$-category" we mean
-a weak $n$-category with strong duality.)
+(As is the case throughout this paper, by ``$n$-category" we implicitly intend some notion of
+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
@@ -46,10 +46,11 @@
 the standard $k$-ball.
 In other words,
 
-\xxpar{Morphisms (preliminary version):}
-{For any $k$-manifold $X$ homeomorphic 
+\begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms}
+For any $k$-manifold $X$ homeomorphic 
 to the standard $k$-ball, we have a set of $k$-morphisms
-$\cC_k(X)$.}
+$\cC_k(X)$.
+\end{preliminary-axiom}
 
 Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the 
 standard $k$-ball.
@@ -64,15 +65,17 @@
 (This will imply ``strong duality", among other things.)
 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 
+\begin{axiom}[Morphisms]
+\label{axiom: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.}
+homeomorphisms to the category of sets and bijections.
+\end{axiom}
+
 
 (Note: We usually omit the subscript $k$.)
 
-We are being deliberately vague about what flavor of manifolds we are considering.
+We are so far  being deliberately vague about what flavor of manifolds we are considering.
 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
 They could be topological or PL or smooth.
 \nn{need to check whether this makes much difference --- see pseudo-isotopy below}
@@ -93,16 +96,18 @@
 boundary of a morphism.
 Morphisms are modeled on balls, so their boundaries are modeled on spheres:
 
-\xxpar{Boundaries (domain and range), part 1:}
-{For each $0 \le k \le n-1$, we have a functor $\cC_k$ from 
+\begin{axiom}[Boundaries (spheres)]
+For each $0 \le k \le n-1$, we have a functor $\cC_k$ from 
 the category of $k$-spheres and 
-homeomorphisms to the category of sets and bijections.}
+homeomorphisms to the category of sets and bijections.
+\end{axiom}
 
 (In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.)
 
-\xxpar{Boundaries, part 2:}
-{For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$.
-These maps, for various $X$, comprise a natural transformation of functors.}
+\begin{axiom}[Boundaries (maps)]
+For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$.
+These maps, for various $X$, comprise a natural transformation of functors.
+\end{axiom}
 
 (Note that the first ``$\bd$" above is part of the data for the category, 
 while the second is the ordinary boundary of manifolds.)
@@ -141,16 +146,17 @@
 That is, given compatible domain and range, we should be able to combine them into
 the full boundary of a morphism:
 
-\xxpar{Domain $+$ range $\to$ boundary:}
-{Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$),
+\begin{axiom}[Boundary from domain and range]
+Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere $(0\le k\le n-1)$,
 $B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere (Figure \ref{blah3}).
 Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the 
 two maps $\bd: \cC(B_i)\to \cC(E)$.
-Then (axiom) we have an injective map
+Then we have an injective map
 \[
 	\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S)
 \]
-which is natural with respect to the actions of homeomorphisms.}
+which is natural with respect to the actions of homeomorphisms.
+\end{axiom}
 
 \begin{figure}[!ht]
 $$
@@ -187,8 +193,8 @@
 In the presence of strong duality, these $k$ distinct compositions are subsumed into 
 one general type of composition which can be in any ``direction".
 
-\xxpar{Composition:}
-{Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
+\begin{axiom}[Composition]
+Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).
 Let $E = \bd Y$, which is a $k{-}2$-sphere.
 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.
@@ -201,14 +207,16 @@
 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
 to the intersection of the boundaries of $B$ and $B_i$.
 If $k < n$ we require that $\gl_Y$ is injective.
-(For $k=n$, see below.)}
+(For $k=n$, see below.)
+\end{axiom}
 
 \begin{figure}[!ht]
 $$\mathfig{.4}{tempkw/blah5}$$
 \caption{From two balls to one ball}\label{blah5}\end{figure}
 
-\xxpar{Strict associativity:}
-{The composition (gluing) maps above are strictly associative.}
+\begin{axiom}[Strict associativity]
+The composition (gluing) maps above are strictly associative.
+\end{axiom}
 
 \begin{figure}[!ht]
 $$\mathfig{.65}{tempkw/blah6}$$
@@ -242,8 +250,8 @@
 
 The next axiom is related to identity morphisms, though that might not be immediately obvious.
 
-\xxpar{Product (identity) morphisms:}
-{Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$.
+\begin{axiom}[Product (identity) morphisms]
+Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$.
 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
 \[ \xymatrix{
@@ -274,7 +282,7 @@
 	\res_{X\times E}(a\times D) = a\times E
 \]
 for $E\sub \bd D$ and $a\in \cC(X)$.
-}
+\end{axiom}
 
 \nn{need even more subaxioms for product morphisms?}
 
@@ -301,10 +309,11 @@
 
 We start with the plain $n$-category case.
 
-\xxpar{Isotopy invariance in dimension $n$ (preliminary version):}
-{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
+\begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$}
+Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
-Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.}
+Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.
+\end{preliminary-axiom}
 
 This axiom needs to be strengthened to force product morphisms to act as the identity.
 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
@@ -333,10 +342,12 @@
 
 The revised axiom is
 
-\xxpar{Extended isotopy invariance in dimension $n$:}
-{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
+\begin{axiom}[Extended isotopy invariance in dimension $n$]
+\label{axiom:extended-isotopies}
+Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity.
-Then $f$ acts trivially on $\cC(X)$.}
+Then $f$ acts trivially on $\cC(X)$.
+\end{axiom}
 
 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
 
@@ -346,8 +357,8 @@
 isotopy invariance with the requirement that families of homeomorphisms act.
 For the moment, assume that our $n$-morphisms are enriched over chain complexes.
 
-\xxpar{Families of homeomorphisms act in dimension $n$.}
-{For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes
+\begin{axiom}[Families of homeomorphisms act in dimension $n$]
+For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes
 \[
 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
 \]
@@ -357,7 +368,8 @@
 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
 a diagram like the one in Proposition \ref{CDprop} commutes.
 \nn{repeat diagram here?}
-\nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}}
+\nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}
+\end{axiom}
 
 We should strengthen the above axiom to apply to families of extended homeomorphisms.
 To do this we need to explain how extended homeomorphisms form a topological space.