--- a/text/intro.tex Fri Jan 07 14:19:50 2011 -0800
+++ b/text/intro.tex Fri Jan 07 14:40:58 2011 -0800
@@ -63,29 +63,30 @@
Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another
definition of an $n$-category, or rather a definition of an $n$-category with strong duality.
(Removing the duality conditions from our definition would make it more complicated rather than less.)
-We call these ``topological $n$-categories'', to differentiate them from previous versions.
+We call these ``disk-like $n$-categories'', to differentiate them from previous versions.
Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms.
+(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 topological $n$-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
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
-topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$.
+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
of sphere modules.
When $n=1$ this just the familiar 2-category of 1-categories, bimodules and intertwinors.
-In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category
+In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a disk-like $n$-category
(using a colimit along certain decompositions of a manifold into balls).
With this in hand, we write $\bc_*(M; \cC)$ to indicate the blob complex of a manifold $M$
with the system of fields constructed from the $n$-category $\cC$.
%\nn{KW: I don't think we use this notational convention any more, right?}
In \S \ref{sec:ainfblob} we give an alternative definition
of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit).
-Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an
+Using these definitions, we show how to use the blob complex to ``resolve" any ordinary $n$-category as an
$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex.
We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants),
in particular the ``gluing formula" of Theorem \ref{thm:gluing} below.
@@ -105,7 +106,7 @@
\newcommand{\yyc}{6}
\node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category};
-\node[box] at (\xxa,\yya) (C) {$\cC$ \\ a topological \\ $n$-category};
+\node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category};
\node[box] at (\xxb,\yya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space};
\node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations};
\node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex};
@@ -148,7 +149,7 @@
make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$,
as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras.
%Appendix \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra,
-%thought of as a topological $n$-category, in terms of the topology of $M$.
+%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)
@@ -371,15 +372,15 @@
for any homeomorphic pair $X$ and $Y$,
satisfying corresponding conditions.
-In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields.
-Below, when we talk about the blob complex for a topological $n$-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 topological $n$-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}}
\begin{ex:blob-complexes-of-balls}[Blob complexes of products with balls form an $A_\infty$ $n$-category]
%\label{thm:blobs-ainfty}
-Let $\cC$ be a topological $n$-category.
+Let $\cC$ be an ordinary $n$-category.
Let $Y$ be an $n{-}k$-manifold.
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$,
to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set
@@ -389,12 +390,12 @@
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 a topological $n$-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 a topological $n$-category; this is described in \S \ref{sec:ainfblob}.
+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 notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.
@@ -412,9 +413,9 @@
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
(see \S \ref{ss:product-formula}).
-Fix a topological $n$-category $\cC$, which we'll omit from the notation.
+Fix a disk-like $n$-category $\cC$, which we'll omit from the notation.
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
-(See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
+(See Appendix \ref{sec:comparing-A-infty} for the translation between disk-like $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.)
\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}}