--- a/text/ncat.tex Wed Jun 29 11:51:35 2011 -0700
+++ b/text/ncat.tex Wed Jun 29 12:02:47 2011 -0700
@@ -34,9 +34,8 @@
The axioms for an $n$-category are spread throughout this section.
Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms},
-\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and
-\ref{axiom:extended-isotopies}.
-For an enriched $n$-category we add \ref{axiom:enriched}.
+\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and \ref{axiom:vcones}.
+For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
For an $A_\infty$ $n$-category, we replace
Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
@@ -579,7 +578,7 @@
Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
(Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.)
-Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which
+Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act
trivially on $\bd b$.
Then $f(b) = b$.
In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially on
@@ -654,7 +653,7 @@
The revised axiom is
%\addtocounter{axiom}{-1}
-\begin{axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$]
+\begin{axiom}[Extended isotopy invariance in dimension $n$]
\label{axiom:extended-isotopies}
Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which
acts trivially on the restriction $\bd b$ of $b$ to $\bd X$.
@@ -876,7 +875,7 @@
or more generally an appropriate sort of $\infty$-category,
we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
to require that families of homeomorphisms act
-and obtain an $A_\infty$ $n$-category.
+and obtain what we shall call an $A_\infty$ $n$-category.
\noop{
We believe that abstract definitions should be guided by diverse collections
@@ -892,7 +891,7 @@
and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$
(e.g.\ the singular chain functor $C_*$).
-\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
+\begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.]
\label{axiom:families}
For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism
\[
@@ -913,12 +912,12 @@
We now describe the topology on $\Coll(X; c)$.
We retain notation from the above definition of collar map.
Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to
-(possibly zero-width) embedded intervals in $X$ terminating at $p$.
+(possibly length zero) embedded intervals in $X$ terminating at $p$.
If $p \in Y$ this interval is the image of $\{p\}\times J$.
-If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$.
+If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$.
Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this.
-Note in particular that parts of the collar are allowed to shrink continuously to zero width.
-(This is the real content; if nothing shrinks to zero width then the action of families of collar
+Note in particular that parts of the collar are allowed to shrink continuously to zero length.
+(This is the real content; if nothing shrinks to zero length then the action of families of collar
maps follows from the action of families of homeomorphisms and compatibility with gluing.)
The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets}
@@ -1118,9 +1117,9 @@
The case $n=d$ captures the $n$-categorical nature of bordisms.
The case $n > 2d$ captures the full symmetric monoidal $n$-category structure.
\end{example}
-\begin{remark}
+\begin{rem}
Working with the smooth bordism category would require careful attention to either collars, corners or halos.
-\end{remark}
+\end{rem}
%\nn{the next example might be an unnecessary distraction. consider deleting it.}
@@ -1344,7 +1343,7 @@
Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$.
Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$.
-We will define $\psi_{\cC;W}(x)$ be be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions
+We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions
related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$.
By Axiom \ref{nca-boundary}, we have a map
\[
@@ -1361,7 +1360,7 @@
along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree
(with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map).
The $i$-th condition is defined similarly.
-Note that these conditions depend on on the boundaries of elements of $\prod_a \cC(X_a)$.
+Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$.
We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the
above conditions for all $i$ and also all
@@ -1440,7 +1439,7 @@
of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$
is permissible.
We can now define the gluing $y_1\bullet y_2$ in the obvious way, and a further application of Axiom \ref{axiom:vcones}
-shows that this is independebt of the choices of representatives of $y_i$.
+shows that this is independent of the choices of representatives of $y_i$.
\medskip
@@ -1454,7 +1453,7 @@
\]
where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation
induced by refinement and gluing.
-If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold,
+If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold,
we can take
\begin{equation*}
\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K,