
diff -r 9fbd8e63ab2e -r c56a3fe75d1e pnas/pnas.tex
--- a/pnas/pnas.tex	Mon Nov 22 09:02:17 2010 -0700
+++ b/pnas/pnas.tex	Mon Nov 22 09:46:07 2010 -0700
@@ -184,14 +184,14 @@
 Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$,
 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$.
 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional
-TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds, 
-but only to mapping cylinders.
+TQFTs, which are slightly weaker structures in that they assign 
+invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds.
 
 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$,
 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$.
 The TQFT gluing rule in dimension $n$ states that
 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$,
-where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$.
+where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$.
 
 When $k=0$ we have an $n$-category $A(pt)$.
 This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$
@@ -207,7 +207,7 @@
 extended all the way down to dimension 0.)
 
 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate.
-For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory
+For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory
 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}.
 Long exact sequences are important computational tools in these theories,
 and also in Khovanov homology, but the colimit construction breaks exactness.
@@ -241,8 +241,8 @@
 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}.
 They are more general in that we make no duality assumptions in the top dimension $n{+}1$.
 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$.
-Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while
-Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs.
+Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while
+Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs.
 
 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. 
 In this paper we attempt to give a clear view of the big picture without getting 
@@ -318,12 +318,12 @@
 Note that the functoriality in the above axiom allows us to operate via
 homeomorphisms which are not the identity on the boundary of the $k$-ball.
 The action of these homeomorphisms gives the ``strong duality" structure.
-As such, we don't subdivide the boundary of a morphism
-into domain and range --- the duality operations can convert between domain and range.
+For this reason we don't subdivide the boundary of a morphism
+into domain and range in the next axiom --- the duality operations can convert between domain and range.
 
 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ 
-from arbitrary manifolds to sets. We need  these functors for $k$-spheres, 
-for $k<n$, for the next axiom.
+defined on arbitrary manifolds. 
+We need  these functors for $k$-spheres, for $k<n$, for the next axiom.
 
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
@@ -374,9 +374,9 @@
 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 $k < n$,
-or if $k=n$ and we are in the $A_\infty$ case \nn{Kevin: remind me why we ask this?}, 
+or if $k=n$ and we are in the $A_\infty$ case, 
 we require that $\gl_Y$ is injective.
-(For $k=n$ in the isotopy $n$-category case, see below. \nn{where?})
+(For $k=n$ in the isotopy $n$-category case, see Axiom \ref{axiom:extended-isotopies}.)
 \end{axiom}
 
 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity}
@@ -385,8 +385,12 @@
 $$\bigsqcup B_i \to B,$$ 
 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result.
 \end{axiom}
-This axiom is only reasonable because the definition assigns a set to every ball; 
-any identifications would limit the extent to which we can demand associativity.
+%This axiom is only reasonable because the definition assigns a set to every ball; 
+%any identifications would limit the extent to which we can demand associativity.
+%%%% KW: It took me quite a while figure out what you [or I??] meant by the above, so I'm attempting a rewrite.
+Note that even though our $n$-categories are ``weak" in the traditional sense, we can require
+strict associativity because we have more morphisms (cf.\ discussion of Moore loops above).
+
 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices.
 \begin{axiom}[Product (identity) morphisms]
 \label{axiom:product}
@@ -486,7 +490,7 @@
 a diagram like the one in Theorem \ref{thm:CH} commutes.
 \end{axiom}
 
-\subsection{Example (the fundamental $n$-groupoid)}
+\subsection{Example (the fundamental $n$-groupoid)} \mbox{}
 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$.
 When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$
 to be the set of continuous maps from $X$ to $T$.
@@ -500,7 +504,7 @@
 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes).
 
 
-\subsection{Example (string diagrams)}
+\subsection{Example (string diagrams)} \mbox{}
 Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).
 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$;
 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$.
@@ -508,7 +512,7 @@
 Boundary restrictions and gluing are again straightforward to define.
 Define product morphisms via product cell decompositions.
 
-\subsection{Example (bordism)}
+\subsection{Example (bordism)} \mbox{}
 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional
 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely
 to $\bd X$.




