
diff -r c06a899bd1f0 -r d3b05641e7ca text/ncat.tex
--- a/text/ncat.tex	Sun Jul 04 13:15:03 2010 -0600
+++ b/text/ncat.tex	Sun Jul 04 23:32:48 2010 -0600
@@ -271,7 +271,7 @@
 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls.
 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from 
 the smaller balls to $X$.
-We  say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'.
+We  say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$".
 In situations where the subdivision is notationally anonymous, we will write
 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)
 the unnamed subdivision.
@@ -667,7 +667,7 @@
 \begin{example}[Maps to a space]
 \rm
 \label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space.
+Fix a ``target space" $T$, any topological space.
 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of 
 all continuous maps from $X$ to $T$.
@@ -704,12 +704,12 @@
 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
 \end{example}
 
-The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend.
-Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
+The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend.
+Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here.
 \begin{example}[Traditional $n$-categories]
 \rm
 \label{ex:traditional-n-categories}
-Given a `traditional $n$-category with strong duality' $C$
+Given a ``traditional $n$-category with strong duality" $C$
 define $\cC(X)$, for $X$ a $k$-ball with $k < n$,
 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}).
 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear
@@ -725,7 +725,7 @@
 to be the dual Hilbert space $A(X\times F; c)$.
 \nn{refer elsewhere for details?}
 
-Recall we described a system of fields and local relations based on a `traditional $n$-category' 
+Recall we described a system of fields and local relations based on a ``traditional $n$-category" 
 $C$ in Example \ref{ex:traditional-n-categories(fields)} above.
 \nn{KW: We already refer to \S \ref{sec:fields} above}
 Constructing a system of fields from $\cC$ recovers that example. 
@@ -794,15 +794,15 @@
 This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
 Notice that with $F$ a point, the above example is a construction turning a topological 
 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
-We think of this as providing a `free resolution' 
-\nn{`cofibrant replacement'?}
+We think of this as providing a ``free resolution" 
+\nn{``cofibrant replacement"?}
 of the topological $n$-category. 
 \todo{Say more here!} 
 In fact, there is also a trivial, but mostly uninteresting, way to do this: 
 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, 
 and take $\CD{B}$ to act trivially. 
 
-Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
+Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
 It's easy to see that with $n=0$, the corresponding system of fields is just 
 linear combinations of connected components of $T$, and the local relations are trivial.
 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -895,12 +895,12 @@
 system of fields and local relations, followed by the usual TQFT definition of a 
 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
-Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', 
+Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", 
 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant 
 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex  for $M$ with coefficients in $\cC$.
 
-We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. 
+We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. 
 An $n$-category $\cC$ provides a functor from this poset to the category of sets, 
 and we  will define $\cC(W)$ as a suitable colimit 
 (or homotopy colimit in the $A_\infty$ case) of this functor. 
@@ -909,7 +909,7 @@
 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).
 
 \begin{defn}
-Say that a `permissible decomposition' of $W$ is a cell decomposition
+Say that a ``permissible decomposition" of $W$ is a cell decomposition
 \[
 	W = \bigcup_a X_a ,
 \]
@@ -938,7 +938,7 @@
 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
 are splittable along this decomposition.
-%For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
+%For a $k$-cell $X$ in a cell composition of $W$, we can consider the ``splittable fields" $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
 
 \begin{defn}
 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
@@ -1740,7 +1740,7 @@
 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side)
 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side)
 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball).
-Corresponding to this decomposition we have a composition (or `gluing') map
+Corresponding to this decomposition we have a composition (or ``gluing") map
 from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$.
 
 \medskip




