
diff -r d8151afc725e -r c861ec0b1554 text/intro.tex
--- 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.
 




