--- a/text/intro.tex Tue Aug 09 23:55:13 2011 -0700
+++ b/text/intro.tex Sun Sep 25 14:33:30 2011 -0600
@@ -64,34 +64,34 @@
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 ``disk-like $n$-categories'', to differentiate them from previous versions.
-Moreover, we find that we need analogous $A_\infty$ disk-like $n$-categories, and we define these as well following very similar axioms.
+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 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 $A_\infty$ disk-like $n$-category, we associate a chain complex instead of a vector space to
+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 $A_\infty$ disk-like $n$-category are designed to capture two main examples:
+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 disk-like $n$-categories can be viewed as objects in a disk-like $n{+}1$-category
+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 intertwiners.
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 disk-like $n$-category $\cC$.
+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 $A_\infty$ disk-like $n$-category on an $n$-manifold (analogously, using a homotopy colimit).
-Using these definitions, we show how to use the blob complex to ``resolve" any ordinary disk-like $n$-category as an
-$A_\infty$ disk-like $n$-category, and relate the first and second definitions of the blob complex.
-We use the blob complex for $A_\infty$ disk-like $n$-categories to establish important properties of the blob complex (in both variants),
+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 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.
The relationship between all these ideas is sketched in Figure \ref{fig:outline}.
@@ -155,8 +155,8 @@
a higher dimensional generalization of the Deligne conjecture
(that the little discs operad acts on Hochschild cochains) in terms of the blob complex.
The appendices prove technical results about $\CH{M}$ and
-make connections between our definitions of disk-like $n$-categories and familiar definitions for $n=1$ and $n=2$,
-as well as relating the $n=1$ case of our $A_\infty$ disk-like $n$-categories with usual $A_\infty$ algebras.
+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 disk-like $n$-category, in terms of the topology of $M$.
@@ -373,42 +373,42 @@
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 $A_\infty$ disk-like $n$-category.
+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 $A_\infty$ disk-like $n$-categories from ordinary disk-like $n$-categories:
+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 $A_\infty$ disk-like $n$-category]
+\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 an ordinary disk-like $n$-category.
+Let $\cC$ be an ordinary $n$-category.
Let $Y$ be an $n{-}k$-manifold.
-There is an $A_\infty$ disk-like $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$,
+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
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.)
-These sets have the structure of an $A_\infty$ disk-like $k$-category, with compositions coming from the gluing map in
+These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in
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 $A_\infty$ disk-like $n$-category from an ordinary disk-like $n$-category.
-We think of this $A_\infty$ disk-like $n$-category as a free resolution.
+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 $A_\infty$ disk-like $n$-category
-instead of an ordinary disk-like $n$-category; this is described in \S \ref{sec:ainfblob}.
+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 $A_\infty$ disk-like $n$-categories constructed as in the previous example.
+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}}
\begin{thm:product}[Product formula]
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
-Let $\cC$ be a disk-like $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ disk-like $k$-category associated to $Y$ via blob homology
+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}).
Then
\[
@@ -420,7 +420,7 @@
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 $A_\infty$ disk-like $1$-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}}
@@ -447,7 +447,7 @@
\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}}
\begin{thm:map-recon}[Mapping spaces]
-Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ disk-like $n$-category based on maps
+Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps
$B^n \to T$.
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)
Then
@@ -512,11 +512,11 @@
since we think of the higher homotopies not as morphisms of the $n$-category but
rather as belonging to some auxiliary category (like chain complexes)
that we are enriching in.
-We have decided to call them ``$A_\infty$ disk-like $n$-categories", since they are a natural generalization
+We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization
of the familiar $A_\infty$ 1-categories.
We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories".
When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense
-we will say ``ordinary disk-like $n$-category".
+we will say ``ordinary $n$-category".
% small problem: our n-cats are of course strictly associative, since we have more morphisms.
% when we say ``associative only up to homotopy" above we are thinking about
% what would happen we we tried to convert to a more traditional n-cat with fewer morphisms