
diff -r d3b05641e7ca -r a96f3d2ef852 text/ncat.tex
--- a/text/ncat.tex	Sun Jul 04 23:32:48 2010 -0600
+++ b/text/ncat.tex	Mon Jul 05 07:47:23 2010 -0600
@@ -663,11 +663,13 @@
 
 
 We now describe several classes of examples of $n$-categories satisfying our axioms.
+We typically specify only the morphisms; the rest of the data for the category
+(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious.
 
 \begin{example}[Maps to a space]
 \rm
 \label{ex:maps-to-a-space}%
-Fix a ``target space" $T$, any topological space.
+Let $T$be a 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 $T$.
@@ -676,10 +678,14 @@
 (Note that homotopy invariance implies isotopy invariance.)
 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
+\end{example}
 
+\noop{
 Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above.
 Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example.
-\end{example}
+\nn{shouldn't this go elsewhere?  we haven't yet discussed constructing a system of fields from
+an n-cat}
+}
 
 \begin{example}[Maps to a space, with a fiber]
 \rm
@@ -701,7 +707,8 @@
 the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$,
 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy
 $h: X\times F\times I \to T$, then $a = \alpha(h)b$.
-\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
+(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices.
+Alternatively, we could equip the balls with fundamental classes.)
 \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.
@@ -723,8 +730,12 @@
 to be the set of all $C$-labeled embedded cell complexes of $X\times F$.
 Define $\cC(X; c)$, for $X$ an $n$-ball,
 to be the dual Hilbert space $A(X\times F; c)$.
-\nn{refer elsewhere for details?}
+(See Subsection \ref{sec:constructing-a-tqft}.)
+\end{example}
 
+\noop{
+\nn{shouldn't this go elsewhere?  we haven't yet discussed constructing a system of fields from
+an n-cat}
 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}
@@ -734,11 +745,8 @@
 where the quotient is built in.
 but (string diagrams)/(relations) is isomorphic to 
 (pasting diagrams composed of smaller string diagrams)/(relations)}
-\end{example}
+}
 
-Finally, we describe a version of the bordism $n$-category suitable to our definitions.
-
-\nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example}
 
 \newcommand{\Bord}{\operatorname{Bord}}
 \begin{example}[The bordism $n$-category, plain version]
@@ -766,15 +774,19 @@
 
 %We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex.
 
-\begin{example}[Chains of maps to a space]
+\begin{example}[Chains (or space) of maps to a space]
 \rm
 \label{ex:chains-of-maps-to-a-space}
 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$.
 For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$.
 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex
-$$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
+\[
+	C_*(\Maps_c(X\times F \to T)),
+\]
+where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
 and $C_*$ denotes singular chains.
-\nn{maybe should also mention version where we enrich over spaces rather than chain complexes}
+Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, 
+we get an $A_\infty$ $n$-category enriched over spaces.
 \end{example}
 
 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to 
@@ -783,7 +795,7 @@
 \begin{example}[Blob complexes of balls (with a fiber)]
 \rm
 \label{ex:blob-complexes-of-balls}
-Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
+Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
 We will define an $A_\infty$ $k$-category $\cC$.
 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
 When $X$ is an $k$-ball,
@@ -795,9 +807,8 @@
 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"?}
 of the topological $n$-category. 
-\todo{Say more here!} 
+\nn{say something about cofibrant replacements?}
 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. 




