--- a/text/ncat.tex Mon Oct 03 16:40:16 2011 -0700
+++ b/text/ncat.tex Tue Oct 04 17:12:08 2011 -0700
@@ -582,6 +582,7 @@
even when we reparametrize our $n$-balls.
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$]
+\label{axiom:isotopy-preliminary}
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.)
@@ -679,7 +680,7 @@
nevertheless we feel that it is too strong.
In the future we would like to see this provisional version of the axiom replaced by something less restrictive.
-We give two alternate versions of he axiom, one better suited for smooth examples, and one better suited to PL examples.
+We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples.
\begin{axiom}[Splittings]
\label{axiom:splittings}
@@ -1003,7 +1004,7 @@
Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the
category $\bbc$ of {\it $n$-balls with boundary conditions}.
Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition".
-The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are
+The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are
homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
%Let $\pi_0(\bbc)$ denote
@@ -1047,7 +1048,7 @@
}
Recall the category $\bbc$ of balls with boundary conditions.
-Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space.
+Note that the morphisms $\Homeo(X;c \to X'; c')$ from $(X, c)$ to $(X', c')$ form a topological space.
Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes)
and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$
(e.g.\ the singular chain functor $C_*$).
@@ -1056,7 +1057,7 @@
\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
\[
- \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') .
+ \cJ(\Homeo(X;c \to X'; c')) \ot \cC(X; c) \to \cC(X'; c') .
\]
Similarly, we have an $\cS$-morphism
\[
@@ -1071,7 +1072,7 @@
\end{axiom}
We now describe the topology on $\Coll(X; c)$.
-We retain notation from the above definition of collar map.
+We retain notation from the above definition of collar map (after Axiom \ref{axiom:isotopy-preliminary}).
Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to
(possibly length zero) embedded intervals in $X$ terminating at $p$.
If $p \in Y$ this interval is the image of $\{p\}\times J$.
@@ -1082,14 +1083,14 @@
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}
-$\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above
-action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible).
+$\Homeo(X;c\to X'; c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above
+action of $\cJ(\Homeo(X;c\to X'; c'))$ to be strictly associative as well (assuming the two actions are compatible).
In fact, compatibility implies less than this.
For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor.
(This is the example most relevant to this paper.)
-Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action
-of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
-And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction.
+Then compatibility implies that the action of $C_*(\Homeo(X;c\to X'; c'))$ agrees with the action
+of $C_0(\Homeo(X;c\to X'; c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero.
+And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction (see Example \ref{ex:blob-complexes-of-balls} below).
Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions,
such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom.
@@ -1142,7 +1143,7 @@
invariance in dimension $n$, while in the fields definition we
instead remember a subspace of local relations which contain differences of isotopic fields.
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.)
-Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
+Thus a \nn{lemma-ize} system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
balls and, at level $n$, quotienting out by the local relations:
\begin{align*}
\cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases}
@@ -1245,7 +1246,7 @@
Let $W$ be an $n{-}j$-manifold.
Define the $j$-category $\cF(W)$ as follows.
If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$.
-If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$,
+If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$,
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$.
\end{example}
@@ -1998,11 +1999,12 @@
\medskip
-We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the
-plain ball case.
+We can define marked pinched products $\pi:E\to M$ of marked balls similarly to the
+plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval.
+\nn{figure, 2 examples}
Note that a marked pinched product can be decomposed into either
two marked pinched products or a plain pinched product and a marked pinched product.
-%\nn{should maybe give figure}
+\nn{should give figure}
\begin{module-axiom}[Product (identity) morphisms]
For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked