--- a/text/ncat.tex Mon Jul 05 07:47:23 2010 -0600
+++ b/text/ncat.tex Wed Jul 07 10:17:21 2010 -0600
@@ -378,7 +378,6 @@
\[
d: \Delta^{k+m}\to\Delta^k .
\]
-In other words, \nn{each point has a neighborhood blah blah...}
(We thank Kevin Costello for suggesting this approach.)
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball,
@@ -518,7 +517,7 @@
We start with the plain $n$-category case.
-\begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}}
+\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.
@@ -592,7 +591,7 @@
The revised axiom is
\addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.}
+\begin{axiom}[\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$.]
\label{axiom:extended-isotopies}
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
@@ -610,7 +609,7 @@
\addtocounter{axiom}{-1}
-\begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.}
+\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
\[
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
@@ -628,7 +627,7 @@
and we can replace the class of all intervals $J$ with intervals contained in $\r$.
Having chains on the space of collar maps act gives rise to coherence maps involving
weak identities.
-We will not pursue this in this draft of the paper.
+We will not pursue this in detail here.
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
into a plain $n$-category (enriched over graded groups).
@@ -916,7 +915,7 @@
and we will define $\cC(W)$ as a suitable colimit
(or homotopy colimit in the $A_\infty$ case) of this functor.
We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-manifolds with boundary data),
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data),
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}
@@ -971,7 +970,7 @@
fix a field on $\bd W$
(i.e. fix an element of the colimit associated to $\bd W$).
-Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
+Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
\begin{defn}[System of fields functor]
\label{def:colim-fields}
@@ -1036,7 +1035,7 @@
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
-\todo{This next sentence is circular: these maps are an axiom, not a consequence of anything. -S} It is easy to see that
+It is easy to see that
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
comprise a natural transformation of functors.
@@ -1338,10 +1337,10 @@
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all
such maps modulo homotopies fixed on $\bdy B \setminus N$.
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}.
+\end{example}
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
-\end{example}
\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}