--- a/text/ncat.tex Tue Sep 21 17:28:14 2010 -0700
+++ b/text/ncat.tex Tue Sep 21 22:39:17 2010 -0700
@@ -829,7 +829,8 @@
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 $\pi^\infty_{\leq n}(T)$.
+Beware 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 $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -903,14 +904,10 @@
\subsection{From balls to manifolds}
\label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we describe how to extend an $n$-category $\cC$ as described above
+In this section we show how to extend an $n$-category $\cC$ as described above
(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
-This extension is a certain colimit, and we've chosen the notation to remind you of this.
-Thus we show that functors $\cC_k$ satisfying the axioms above have a canonical extension
-from $k$-balls to arbitrary $k$-manifolds.
-Recall that we've already anticipated this construction in the previous section,
-inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls,
-so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
+This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
+
In the case of plain $n$-categories, this construction factors into a construction of a
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}.
@@ -920,7 +917,13 @@
(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 $\cC$.
+same as the original blob complex for $M$ with coefficients in $\cC$.
+
+Recall that we've already anticipated this construction in the previous section,
+inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls,
+so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
+
+\medskip
We will first define the ``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,
@@ -1085,7 +1088,8 @@
\medskip
-$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
+$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
+Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
It is easy to see that
there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps
@@ -1141,8 +1145,8 @@
But this implies that $a = a_0 = a_k = \hat{a}$, contrary to our assumption that $a\ne \hat{a}$.
\end{proof}
-\nn{need to finish explaining why we have a system of fields;
-define $k$-cat $\cC(\cdot\times W)$}
+%\nn{need to finish explaining why we have a system of fields;
+%define $k$-cat $\cC(\cdot\times W)$}
\subsection{Modules}
@@ -2227,7 +2231,7 @@
It is easy to show that this is independent of the choice of $E$.
Note also that this map depends only on the restriction of $f$ to $\bd X$.
In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by
-Axiom \ref{axiom:extended-isotopies} of \S\ref{ss:n-cat-def}.
+Axiom \ref{axiom:extended-isotopies}.
We define product $n{+}1$-morphisms to be identity maps of modules.