
diff -r 28592849a474 -r 9fbd8e63ab2e pnas/pnas.tex
--- a/pnas/pnas.tex	Sun Nov 21 15:24:53 2010 -0800
+++ b/pnas/pnas.tex	Mon Nov 22 09:02:17 2010 -0700
@@ -137,7 +137,7 @@
 
 \begin{abstract}
 We explain the need for new axioms for topological quantum field theories that include ideas from derived 
-categories and homotopy theory. We summarize our axioms for higher categories, and describe the `blob complex'. 
+categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex". 
 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. 
 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$. 
 The higher homology groups should be viewed as generalizations of Hochschild homology. 
@@ -271,10 +271,10 @@
 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
 
 We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.
-These variations are `isotopy $n$-categories', where homeomorphisms fixing the boundary
+These variations are ``isotopy $n$-categories", where homeomorphisms fixing the boundary
 act trivially on the sets associated to $n$-balls
 (and these sets are usually vector spaces or more generally modules over a commutative ring)
-and `$A_\infty$ $n$-categories',  where there is a homotopy action of
+and ``$A_\infty$ $n$-categories",  where there is a homotopy action of
 $k$-parameter families of homeomorphisms on these sets
 (which are usually chain complexes or topological spaces).
 
@@ -339,7 +339,7 @@
 compatible with the $\cS$ structure on $\cC_n(X; c)$.
 
 
-Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to 
+Given two hemispheres (a ``domain" and ``range") that agree on the equator, we need to be able to 
 assemble them into a boundary value of the entire sphere.
 
 \begin{lem}
@@ -385,7 +385,8 @@
 $$\bigsqcup B_i \to B,$$ 
 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.
 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}
@@ -500,7 +501,7 @@
 
 
 \subsection{Example (string diagrams)}
-Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).
+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 $C$.
 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$.
@@ -562,12 +563,12 @@
 \end{equation*}
 where the restrictions to the various pieces of shared boundaries amongst the balls
 $X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds). 
-When $k=n$, the `subset' and `product' in the above formula should be 
+When $k=n$, the ``subset" and ``product" in the above formula should be 
 interpreted in the appropriate enriching category.
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
 \end{defn}
 
-We will use the term `field on $W$' to refer to a point of this functor,
+We will use the term ``field on $W$" to refer to a point of this functor,
 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 
 
@@ -607,9 +608,10 @@
 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, 
 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, 
 and taking product identifies the roots of several trees. 
-The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. 
+The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. 
 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required.
-A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product.
+A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. 
+In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product.
 
 %When $\cC$ is a topological $n$-category,
 %the flexibility available in the construction of a homotopy colimit allows
@@ -638,7 +640,7 @@
 \item for each resulting piece of $W$, a field,
 \end{itemize}
 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. 
-We call such a field a `null field on $B$'.
+We call such a field a ``null field on $B$".
 
 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs.
 
@@ -808,7 +810,7 @@
 family of homeomorphisms can be localized to at most $k$ small sets.
 
 With this alternate version in hand, the theorem is straightforward.
-By functoriality (Property \cite{property:functoriality}) $Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this
+By functoriality (Property \ref{property:functoriality}) $\Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this
 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$
 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$.
 It is easy to check that $e_X$ thus defined has the desired properties.




