--- a/text/comm_alg.tex	Tue Jun 01 21:44:09 2010 -0700
+++ b/text/comm_alg.tex	Tue Jun 01 23:07:42 2010 -0700
@@ -109,8 +109,8 @@
 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
 Let us check this directly.
 
-According to \cite[3.2.2]{MR1600246}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$.
-\nn{say something about $t$-degree?  is this in Loday?}
+The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
+(See also  \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
 
 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other.
 The fixed points of this flow are the equally spaced configurations.
@@ -123,9 +123,9 @@
 of course $\Sigma^0(S^1)$ is a point.
 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$
 and is zero for $i\ge 2$.
-\nn{say something about $t$-degrees also matching up?}
+Note that the $j$-grading here matches with the $t$-grading on the algebraic side.
 
-By xxxx and \ref{ktchprop}, 
+By xxxx and Proposition \ref{ktchprop}, 
 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$.
 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$.
 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree

--- a/text/deligne.tex	Tue Jun 01 21:44:09 2010 -0700
+++ b/text/deligne.tex	Tue Jun 01 23:07:42 2010 -0700
@@ -11,7 +11,7 @@
 (Proposition \ref{prop:deligne} below).
 Then we sketch the proof.
 
-\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S}
+\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S}
 
 %from http://www.ams.org/mathscinet-getitem?mr=1805894
 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. 

--- a/text/intro.tex	Tue Jun 01 21:44:09 2010 -0700
+++ b/text/intro.tex	Tue Jun 01 23:07:42 2010 -0700
@@ -24,12 +24,12 @@
 
 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 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.
+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 (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we 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.
 
 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 groupoid.
 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 (using a colimit along cellulations of a manifold), and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). 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{ss:ncat_fields}  we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). 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.
 
 The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
 
@@ -48,11 +48,11 @@
 \newcommand{\yc}{6}
 
 \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category};
-\node[box] at (\xb,\ya) (A) {$A(M; \cC)$ \\ the (dual) TQFT \\ Hilbert space};
+\node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space};
 \node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations};
 \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex};
 \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category};
-\node[box] at (\xb,\yc) (BCs) {$\bc_*(M; \cC_*)$};
+\node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$};
 
 
 
@@ -77,7 +77,7 @@
 \label{fig:outline}
 \end{figure}
 
-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 generalization 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 familiar 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 generalization 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 the `small blob complex', and make connections between our definitions of $n$-categories and familiar 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}
@@ -348,7 +348,7 @@
 
 
 \subsection{Thanks and acknowledgements}
-We'd like to thank David Ben-Zvi, Kevin Costello, 
+We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas,
 Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. 
 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.
 

--- a/text/ncat.tex	Tue Jun 01 21:44:09 2010 -0700
+++ b/text/ncat.tex	Tue Jun 01 23:07:42 2010 -0700
@@ -74,7 +74,7 @@
 We will concentrate on the case of PL unoriented manifolds.
 
 (The ambitious reader may want to keep in mind two other classes of balls.
-The first is balls equipped with a map to some other space $Y$. \todo{cite something of Teichner's?}
+The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). 
 This will be used below to describe the blob complex of a fiber bundle with
 base space $Y$.
 The second is balls equipped with a section of the the tangent bundle, or the frame
@@ -86,7 +86,7 @@
 of morphisms).
 The 0-sphere is unusual among spheres in that it is disconnected.
 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
-(Actually, this is only true in the oriented case, with 1-morphsims parameterized
+(Actually, this is only true in the oriented case, with 1-morphisms parameterized
 by oriented 1-balls.)
 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place.
 
@@ -123,7 +123,7 @@
 
 Most of the examples of $n$-categories we are interested in are enriched in the following sense.
 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
-all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category
+all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category
 (e.g.\ vector spaces, or modules over some ring, or chain complexes),
 and all the structure maps of the $n$-category should be compatible with the auxiliary
 category structure.
@@ -142,7 +142,7 @@
 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].)
+(cf. \cite{MR2079378}.)
 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.}
@@ -780,23 +780,6 @@
 (actions of homeomorphisms);
 define $k$-cat $\cC(\cdot\times W)$}
 
-\nn{need to revise stuff below, since we no longer have the sphere axiom}
-
-Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction.
-
-\begin{lem}
-For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$
-\end{lem}
-
-\begin{lem}
-For a topological $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) = \cC(B).$$
-\end{lem}
-
-\begin{lem}
-For an $A_\infty$ $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) \quism \cC(B).$$
-\end{lem}
-
-
 \subsection{Modules}
 
 Next we define plain and $A_\infty$ $n$-category modules.

--- a/text/tqftreview.tex	Tue Jun 01 21:44:09 2010 -0700
+++ b/text/tqftreview.tex	Tue Jun 01 23:07:42 2010 -0700
@@ -4,8 +4,8 @@
 \label{sec:fields}
 \label{sec:tqftsviafields}
 
-In this section we review the construction of TQFTs from ``topological fields".
-For more details see \cite{kw:tqft}.
+In this section we review the notion of a ``system of fields and local relations".
+For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-TQFT}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations.
 
 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
 submanifold of $X$, then $X \setmin Y$ implicitly means the closure
@@ -21,18 +21,17 @@
 oriented, topological, smooth, spin, etc. --- but for definiteness we
 will stick with unoriented PL.)
 
-%Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.
+Fix a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$.
 
 A $n$-dimensional {\it system of fields} in $\cS$
 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$
 together with some additional data and satisfying some additional conditions, all specified below.
 
-Before finishing the definition of fields, we give two motivating examples
-(actually, families of examples) of systems of fields.
+Before finishing the definition of fields, we give two motivating examples of systems of fields.
 
 \begin{example}
 \label{ex:maps-to-a-space(fields)}
-Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps
+Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps
 from X to $B$.
 \end{example}
 
@@ -42,7 +41,7 @@
 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by
 $j$-morphisms of $C$.
 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.
-This is described in more detail below.
+This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}.
 \end{example}
 
 Now for the rest of the definition of system of fields.
@@ -144,6 +143,47 @@
 \nn{remark that if top dimensional fields are not already linear
 then we will soon linearize them(?)}
 
+For top dimensional ($n$-dimensional) manifolds, we're actually interested
+in the linearized space of fields.
+By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
+the vector space of finite
+linear combinations of fields on $X$.
+If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
+Thus the restriction (to boundary) maps are well defined because we never
+take linear combinations of fields with differing boundary conditions.
+
+In some cases we don't linearize the default way; instead we take the
+spaces $\lf(X; a)$ to be part of the data for the system of fields.
+In particular, for fields based on linear $n$-category pictures we linearize as follows.
+Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
+obvious relations on 0-cell labels.
+More specifically, let $L$ be a cell decomposition of $X$
+and let $p$ be a 0-cell of $L$.
+Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
+$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
+Then the subspace $K$ is generated by things of the form
+$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
+to infer the meaning of $\alpha_{\lambda c + d}$.
+Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
+
+\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
+will do something similar below; in general, whenever a label lives in a linear
+space we do something like this; ? say something about tensor
+product of all the linear label spaces?  Yes:}
+
+For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
+Define an ``almost-field" to be a field without labels on the 0-cells.
+(Recall that 0-cells are labeled by $n$-morphisms.)
+To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
+space determined by the labeling of the link of the 0-cell.
+(If the 0-cell were labeled, the label would live in this space.)
+We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
+We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
+above tensor products.
+
+
+\subsection{Systems of fields from $n$-categories}
+\label{sec:example:traditional-n-categories(fields)}
 We now describe in more detail systems of fields coming from sub-cell-complexes labeled
 by $n$-category morphisms.
 
@@ -226,43 +266,6 @@
 
 \medskip
 
-For top dimensional ($n$-dimensional) manifolds, we're actually interested
-in the linearized space of fields.
-By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
-the vector space of finite
-linear combinations of fields on $X$.
-If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
-Thus the restriction (to boundary) maps are well defined because we never
-take linear combinations of fields with differing boundary conditions.
-
-In some cases we don't linearize the default way; instead we take the
-spaces $\lf(X; a)$ to be part of the data for the system of fields.
-In particular, for fields based on linear $n$-category pictures we linearize as follows.
-Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
-obvious relations on 0-cell labels.
-More specifically, let $L$ be a cell decomposition of $X$
-and let $p$ be a 0-cell of $L$.
-Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
-$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
-Then the subspace $K$ is generated by things of the form
-$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
-to infer the meaning of $\alpha_{\lambda c + d}$.
-Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
-
-\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
-will do something similar below; in general, whenever a label lives in a linear
-space we do something like this; ? say something about tensor
-product of all the linear label spaces?  Yes:}
-
-For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
-Define an ``almost-field" to be a field without labels on the 0-cells.
-(Recall that 0-cells are labeled by $n$-morphisms.)
-To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
-space determined by the labeling of the link of the 0-cell.
-(If the 0-cell were labeled, the label would live in this space.)
-We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
-We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
-above tensor products.