--- a/text/intro.tex Fri May 06 18:02:06 2011 -0700
+++ b/text/intro.tex Sat May 07 08:35:36 2011 -0700
@@ -68,11 +68,14 @@
(See \S \ref{n-cat-names} below for a discussion of $n$-category terminology.)
The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms.
-We try to be as lax as possible: a disk-like $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball.
+We try to be as lax as possible: a disk-like $n$-category associates a
+vector space to every $B$ homeomorphic to the $n$-ball.
These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid.
-For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of
+For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to
+each such $B$ and ask that the action of
homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms.
-The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a
+The axioms for an $A_\infty$ $n$-category are designed to capture two main examples:
+the blob complexes of $n$-balls labelled by a
disk-like $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
In \S \ref{ssec:spherecat} we explain how $n$-categories can be viewed as objects in an $n{+}1$-category
@@ -151,14 +154,6 @@
%Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra,
%thought of as a disk-like $n$-category, in terms of the topology of $M$.
-%%%% this is said later in the intro
-%Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes)
-%even when we could work in greater generality (symmetric monoidal categories, model categories, etc.).
-
-%\item ? one of the points we make (far) below is that there is not really much
-%difference between (a) systems of fields and local relations and (b) $n$-cats;
-%thus we tend to switch between talking in terms of one or the other
-
\subsection{Motivations}
@@ -259,10 +254,6 @@
Note that this includes the case of gluing two disjoint manifolds together.
\begin{property}[Gluing map]
\label{property:gluing-map}%
-%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map
-%\begin{equation*}
-%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).
-%\end{equation*}
Given a gluing $X \to X_\mathrm{gl}$, there is
a natural map
\[
@@ -372,9 +363,13 @@
for any homeomorphic pair $X$ and $Y$,
satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories, from which we can construct systems of fields.
-Below, when we talk about the blob complex for a disk-like $n$-category, we are implicitly passing first to this associated system of fields.
-Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. In that section we describe how to use the blob complex to construct $A_\infty$ $n$-categories from ordinary $n$-categories:
+In \S \ref{sec:ncats} we introduce the notion of disk-like $n$-categories,
+from which we can construct systems of fields.
+Below, when we talk about the blob complex for a disk-like $n$-category,
+we are implicitly passing first to this associated system of fields.
+Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category.
+In that section we describe how to use the blob complex to
+construct $A_\infty$ $n$-categories from ordinary $n$-categories:
\newtheorem*{ex:blob-complexes-of-balls}{Example \ref{ex:blob-complexes-of-balls}}
@@ -390,13 +385,16 @@
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.
\end{ex:blob-complexes-of-balls}
\begin{rem}
-Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category.
+Perhaps the most interesting case is when $Y$ is just a point;
+then we have a way of building an $A_\infty$ $n$-category from an ordinary $n$-category.
We think of this $A_\infty$ $n$-category as a free resolution.
\end{rem}
There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category
instead of an ordinary $n$-category; this is described in \S \ref{sec:ainfblob}.
-The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example.
+The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$.
+The next theorem describes the blob complex for product manifolds,
+in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example.
%The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.
\newtheorem*{thm:product}{Theorem \ref{thm:product}}
@@ -404,7 +402,8 @@
\begin{thm:product}[Product formula]
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}).
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology
+(see Example \ref{ex:blob-complexes-of-balls}).
Then
\[
\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).
@@ -532,7 +531,9 @@
by ``disk-like".
(But beware: disks can come in various flavors, and some of them, such as framed disks,
don't actually imply much duality.)
-Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. It will thus be a theorem that our disk-like 2-categories are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}.
+Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense.
+It will thus be a theorem that our disk-like 2-categories
+are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}.
Finally, we need a general name for isomorphisms between balls, where the balls could be
piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof.
@@ -566,7 +567,8 @@
Ansgar Schneider,
and
Dan Berwick-Evans.
-\nn{need to double-check this list once the reading course is over}
-During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive
+During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at
+Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.
+We'd like to thank the Aspen Center for Physics for the pleasant and productive
environment provided there during the final preparation of this manuscript.