# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1316982810 21600
# Node ID a0fd6e6209260dd5f43ebef3f2925699a01a8060
# Parent  7abe7642265e0868e297eb2bf6e3e5160f74e445
Backed out changeset 7abe7642265e

diff -r 7abe7642265e -r a0fd6e620926 RefereeReport.pdf
Binary file RefereeReport.pdf has changed
diff -r 7abe7642265e -r a0fd6e620926 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Tue Aug 09 23:55:13 2011 -0700
+++ b/text/a_inf_blob.tex	Sun Sep 25 14:33:30 2011 -0600
@@ -1,8 +1,8 @@
 %!TEX root = ../blob1.tex
 
-\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} disk-like \texorpdfstring{$n$}{n}-categories}
+\section{The blob complex for \texorpdfstring{$A_\infty$}{A-infinity} \texorpdfstring{$n$}{n}-categories}
 \label{sec:ainfblob}
-Given an $A_\infty$ disk-like $n$-category $\cC$ and an $n$-manifold $M$, we make the 
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the 
 anticlimactically tautological definition of the blob
 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}.
 
@@ -32,7 +32,7 @@
 
 
 Given an $n$-dimensional system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from 
-Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ disk-like $k$-category $\cC_F$ 
+Example \ref{ex:blob-complexes-of-balls} that there is an  $A_\infty$ $k$-category $\cC_F$ 
 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and
 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$.
 
@@ -219,11 +219,11 @@
 
 %\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
 
-If $Y$ has dimension $k-m$, then we have a disk-like $m$-category $\cC_{Y\times F}$ whose value at
+If $Y$ has dimension $k-m$, then we have an $m$-category $\cC_{Y\times F}$ whose value at
 a $j$-ball $X$ is either $\cE(X\times Y\times F)$ (if $j<m$) or $\bc_*(X\times Y\times F)$
 (if $j=m$).
 (See Example \ref{ex:blob-complexes-of-balls}.)
-Similarly we have a disk-like  $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
+Similarly we have an $m$-category whose value at $X$ is $\cl{\cC_F}(X\times Y)$.
 These two categories are equivalent, but since we do not define functors between
 disk-like $n$-categories in this paper we are unable to say precisely
 what ``equivalent" means in this context.
@@ -235,7 +235,7 @@
 
 \begin{cor}
 \label{cor:new-old}
-Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$ disk-like 
+Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$
 $n$-category obtained from $\cE$ by taking the blob complex of balls.
 Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are
 homotopy equivalent:
@@ -261,18 +261,18 @@
 We can generalize the definition of a $k$-category by replacing the categories
 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$
 (c.f. \cite{MR2079378}).
-Call this a disk-like $k$-category over $Y$.
-A fiber bundle $F\to E\to Y$ gives an example of a disk-like $k$-category over $Y$:
+Call this a $k$-category over $Y$.
+A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$,
 or the fields $\cE(p^*(E))$, if $\dim(D) < k$.
 (Here $p^*(E)$ denotes the pull-back bundle over $D$.)
-Let $\cF_E$ denote this disk-like $k$-category over $Y$.
+Let $\cF_E$ denote this $k$-category over $Y$.
 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
 get a chain complex $\cl{\cF_E}(Y)$.
 The proof of Theorem \ref{thm:product} goes through essentially unchanged 
 to show the following result.
 \begin{thm}
-Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the disk-like $k$-category over $Y$ defined above.
+Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.
 Then
 \[
 	\bc_*(E) \simeq \cl{\cF_E}(Y) .
@@ -287,13 +287,13 @@
 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$.
 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$
 lying above $D$.)
-We can define a disk-like $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
+We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
 We can again adapt the homotopy colimit construction to
 get a chain complex $\cl{\cF_M}(Y)$.
 The proof of Theorem \ref{thm:product} again goes through essentially unchanged 
 to show that
 \begin{thm}
-Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the disk-like $k$-category over $Y$ defined above.
+Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above.
 Then
 \[
 	\bc_*(M) \simeq \cl{\cF_M}(Y) .
@@ -315,7 +315,7 @@
 such that the restriction of $E$ to $X_i$ is homeomorphic to a product $F\times X_i$,
 and choose trivializations of these products as well.
 
-Let $\cF$ be the disk-like $k$-category associated to $F$.
+Let $\cF$ be the $k$-category associated to $F$.
 To each codimension-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module) for $\cF$.
 More generally, to each codimension-$m$ face we have an $S^{m-1}$-module for a $(k{-}m{+}1)$-category
 associated to the (decorated) link of that face.
@@ -341,22 +341,22 @@
 $X = X_1\cup (Y\times J) \cup X_2$.
 Given this data we have:
 \begin{itemize}
-\item An $A_\infty$ disk-like $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball
+\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball
 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$
 (for $m+k = n$).
 (See Example \ref{ex:blob-complexes-of-balls}.)
 %\nn{need to explain $c$}.
-\item An $A_\infty$ disk-like $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
+\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked
 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)
 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).
 (See Example \ref{bc-module-example}.)
 \item The tensor product $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$, which is
-an $A_\infty$ disk-like $n{-}k$-category.
+an $A_\infty$ $n{-}k$-category.
 (See \S \ref{moddecss}.)
 \end{itemize}
 
-It is the case that the disk-like $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$
+It is the case that the $n{-}k$-categories $\bc(X)$ and $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$
 are equivalent for all $k$, but since we do not develop a definition of functor between $n$-categories
 in this paper, we cannot state this precisely.
 (It will appear in a future paper.)
@@ -403,7 +403,7 @@
 
 The next theorem shows how to reconstruct a mapping space from local data.
 Let $T$ be a topological space, let $M$ be an $n$-manifold, 
-and recall the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$ 
+and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ 
 of Example \ref{ex:chains-of-maps-to-a-space}.
 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever
 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$).
diff -r 7abe7642265e -r a0fd6e620926 text/appendixes/comparing_defs.tex
--- a/text/appendixes/comparing_defs.tex	Tue Aug 09 23:55:13 2011 -0700
+++ b/text/appendixes/comparing_defs.tex	Sun Sep 25 14:33:30 2011 -0600
@@ -4,8 +4,8 @@
 \label{sec:comparing-defs}
 
 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct
-a disk-like  $n$-category from a traditional $n$-category; the morphisms of the 
-disk-like  $n$-category are string diagrams labeled by the traditional $n$-category.
+a topological $n$-category from a traditional $n$-category; the morphisms of the 
+topological $n$-category are string diagrams labeled by the traditional $n$-category.
 In this appendix we sketch how to go the other direction, for $n=1$ and 2.
 The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms
 of the corresponding traditional $n$-category to be $\cC(B^k)$, where
@@ -575,11 +575,11 @@
 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories}
 \label{sec:comparing-A-infty}
 In this section, we make contact between the usual definition of an $A_\infty$ category 
-and our definition of an $A_\infty$ disk-like $1$-category, from \S \ref{ss:n-cat-def}.
+and our definition of a disk-like $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}.
 
 \medskip
 
-Given an $A_\infty$ disk-like $1$-category $\cC$, we define an ``$m_k$-style" 
+Given a disk-like $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" 
 $A_\infty$ $1$-category $A$ as follows.
 The objects of $A$ are $\cC(pt)$.
 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$
@@ -621,7 +621,7 @@
 Operad associativity for $A$ implies that this gluing map is independent of the choice of
 $g$ and the choice of representative $(f_i, a_i)$.
 
-It is straightforward to verify the remaining axioms for a $A_\infty$ disk-like 1-category.
+It is straightforward to verify the remaining axioms for a disk-like $A_\infty$ 1-category.
 
 
 
diff -r 7abe7642265e -r a0fd6e620926 text/intro.tex
--- 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
diff -r 7abe7642265e -r a0fd6e620926 text/ncat.tex
--- a/text/ncat.tex	Tue Aug 09 23:55:13 2011 -0700
+++ b/text/ncat.tex	Sun Sep 25 14:33:30 2011 -0600
@@ -3,10 +3,10 @@
 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip}
 
-\section{Disk-like \texorpdfstring{$n$}{n}-categories and their modules}
+\section{\texorpdfstring{$n$}{n}-categories and their modules}
 \label{sec:ncats}
 
-\subsection{Definition of disk-like \texorpdfstring{$n$}{n}-categories}
+\subsection{Definition of \texorpdfstring{$n$}{n}-categories}
 \label{ss:n-cat-def}
 
 Before proceeding, we need more appropriate definitions of $n$-categories, 
@@ -32,11 +32,11 @@
 
 \medskip
 
-The axioms for a disk-like $n$-category are spread throughout this section.
-Collecting these together, a disk-like $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, 
+The axioms for an $n$-category are spread throughout this section.
+Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, 
 \ref{nca-boundary}, \ref{axiom:composition},  \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:extended-isotopies} and  \ref{axiom:vcones}.
-For an enriched disk-like $n$-category we add Axiom \ref{axiom:enriched}.
-For an $A_\infty$ disk-like $n$-category, we replace 
+For an enriched $n$-category we add Axiom \ref{axiom:enriched}.
+For an $A_\infty$ $n$-category, we replace 
 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.
 
 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms 
@@ -88,7 +88,7 @@
 %\nn{need to check whether this makes much difference}
 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
 to be fussier about corners and boundaries.)
-For each flavor of manifold there is a corresponding flavor of disk-like $n$-category.
+For each flavor of manifold there is a corresponding flavor of $n$-category.
 For simplicity, we will concentrate on the case of PL unoriented manifolds.
 
 An ambitious reader may want to keep in mind two other classes of balls.
@@ -807,8 +807,8 @@
 
 \medskip
 
-This completes the definition of a disk-like $n$-category.
-Next we define enriched disk-like $n$-categories.
+This completes the definition of an $n$-category.
+Next we define enriched $n$-categories.
 
 \medskip
 
@@ -837,7 +837,7 @@
 For convenience, we will also assume that the objects of our auxiliary category are sets with extra structure.
 (Otherwise, stating the axioms for identity morphisms becomes more cumbersome.)
 
-Before stating the revised axioms for a disk-like $n$-category enriched in a distributive monoidal category,
+Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category,
 we need a preliminary definition.
 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}.
@@ -846,10 +846,10 @@
 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$.
 %Let $\pi_0(\bbc)$ denote
  
-\begin{axiom}[Enriched disk-like $n$-categories]
+\begin{axiom}[Enriched $n$-categories]
 \label{axiom:enriched}
 Let $\cS$ be a distributive symmetric monoidal category.
-A disk-like $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
+An $n$-category enriched in $\cS$ satisfies the above $n$-category axioms for $k=0,\ldots,n-1$,
 and modifies the axioms for $k=n$ as follows:
 \begin{itemize}
 \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$.
@@ -875,7 +875,7 @@
 or more generally an appropriate sort of $\infty$-category,
 we can modify the extended isotopy axiom \ref{axiom:extended-isotopies}
 to require that families of homeomorphisms act
-and obtain what we shall call an $A_\infty$ disk-like $n$-category.
+and obtain what we shall call an $A_\infty$ $n$-category.
 
 \noop{
 We believe that abstract definitions should be guided by diverse collections
@@ -928,7 +928,7 @@
 (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 $A_\infty$ disk-like $n$-category based on the blob construction.
+And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction.
 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.
 
@@ -950,7 +950,7 @@
 For future reference we make the following definition.
 
 \begin{defn}
-A {\em strict $A_\infty$ disk-like $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
+A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative.
 \end{defn}
 
 \noop{
@@ -966,13 +966,13 @@
 
 \medskip
 
-We define a $j$ times monoidal disk-like $n$-category to be a disk-like $(n{+}j)$-category $\cC$ where
+We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where
 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$.
 See Example \ref{ex:bord-cat}.
 
 \medskip
 
-The alert reader will have already noticed that our definition of an (ordinary) disk-like $n$-category
+The alert reader will have already noticed that our definition of an (ordinary) $n$-category
 is extremely similar to our definition of a system of fields.
 There are two differences.
 First, for the $n$-category definition we restrict our attention to balls
@@ -981,7 +981,7 @@
 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 a disk-like $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to
+Thus a 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}
@@ -995,7 +995,7 @@
 In the $n$-category axioms above we have intermingled data and properties for expository reasons.
 Here's a summary of the definition which segregates the data from the properties.
 
-A disk-like $n$-category consists of the following data:
+An $n$-category consists of the following data:
 \begin{itemize}
 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});
@@ -1021,7 +1021,7 @@
 \end{itemize}
 
 
-\subsection{Examples of disk-like \texorpdfstring{$n$}{n}-categories}
+\subsection{Examples of \texorpdfstring{$n$}{n}-categories}
 \label{ss:ncat-examples}
 
 
@@ -1153,7 +1153,7 @@
 where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
 and $C_*$ denotes singular chains.
 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X \to T)$, 
-we get an $A_\infty$ disk-like $n$-category enriched over spaces.
+we get an $A_\infty$ $n$-category enriched over spaces.
 \end{example}
 
 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to 
@@ -1163,7 +1163,7 @@
 \rm
 \label{ex:blob-complexes-of-balls}
 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$.
-We will define an $A_\infty$ disk-like $k$-category $\cC$.
+We will define an $A_\infty$ $k$-category $\cC$.
 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$.
 When $X$ is an $k$-ball,
 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
@@ -1171,17 +1171,17 @@
 \end{example}
 
 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
-Notice that with $F$ a point, the above example is a construction turning an ordinary disk-like
-$n$-category $\cC$ into an $A_\infty$ disk-like $n$-category.
+Notice that with $F$ a point, the above example is a construction turning an ordinary 
+$n$-category $\cC$ into an $A_\infty$ $n$-category.
 We think of this as providing a ``free resolution" 
-of the ordinary disk-like $n$-category. 
+of the ordinary $n$-category. 
 %\nn{say something about cofibrant replacements?}
 In fact, there is also a trivial, but mostly uninteresting, way to do this: 
 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, 
 and take $\CD{B}$ to act trivially. 
 
-Beware that the ``free resolution" of the ordinary disk-like $n$-category $\pi_{\leq n}(T)$ 
-is not the $A_\infty$ disk-like $n$-category $\pi^\infty_{\leq n}(T)$.
+Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ 
+is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
 It's easy to see that with $n=0$, the corresponding system of fields is just 
 linear combinations of connected components of $T$, and the local relations are trivial.
 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
@@ -1225,7 +1225,7 @@
 Let $A$ be an $\cE\cB_n$-algebra.
 Note that this implies a $\Diff(B^n)$ action on $A$, 
 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$.
-We will define a strict $A_\infty$ disk-like $n$-category $\cC^A$.
+We will define a strict $A_\infty$ $n$-category $\cC^A$.
 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point.
 In other words, the $k$-morphisms are trivial for $k<n$.
 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction.
@@ -1237,12 +1237,12 @@
 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$.
 Alternatively and more simply, we could define $\cC^A(X)$ to be 
 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$.
-The remaining data for the $A_\infty$ disk-like $n$-category 
+The remaining data for the $A_\infty$ $n$-category 
 --- composition and $\Diff(X\to X')$ action ---
 also comes from the $\cE\cB_n$ action on $A$.
 %\nn{should we spell this out?}
 
-Conversely, one can show that a strict $A_\infty$  disk-like $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to 
 an $\cE\cB_n$-algebra.
 %\nn{The paper is already long; is it worth giving details here?}
@@ -1257,19 +1257,19 @@
 
 \subsection{From balls to manifolds}
 \label{ss:ncat_fields} \label{ss:ncat-coend}
-In this section we show how to extend a disk-like $n$-category $\cC$ as described above 
+In this section we show how to extend an $n$-category $\cC$ as described above 
 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$.
 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this.
 
-In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a 
+In the case of ordinary $n$-categories, this construction factors into a construction of a 
 system of fields and local relations, followed by the usual TQFT definition of a 
 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
-For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
-Recall that we can take a ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution", 
-an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls 
+For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
+Recall that we can take a ordinary $n$-category $\cC$ and pass to the ``free resolution", 
+an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls 
 (recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant 
-for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the 
+for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the 
 same as the original blob complex for $M$ with coefficients in $\cC$.
 
 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, 
@@ -1279,11 +1279,11 @@
 \medskip
 
 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. 
-A disk-like $n$-category $\cC$ provides a functor from this poset to the category of sets, 
+An $n$-category $\cC$ provides a functor from this poset to the category of sets, 
 and we  will define $\cl{\cC}(W)$ as a suitable colimit 
 (or homotopy colimit in the $A_\infty$ case) of this functor. 
 We'll later give a more explicit description of this colimit.
-In the case that the disk-like $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain 
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain 
 complexes to $n$-balls with boundary data), 
 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into 
 subsets according to boundary data, and each of these subsets has the appropriate structure 
@@ -1334,7 +1334,7 @@
 \label{partofJfig}
 \end{figure}
 
-A disk-like $n$-category $\cC$ determines 
+An $n$-category $\cC$ determines 
 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets 
 (possibly with additional structure if $k=n$).
 Let $x = \{X_a\}$ be a permissible decomposition of $W$ (i.e.\ object of $\cD(W)$).
@@ -1402,14 +1402,14 @@
 
 \begin{defn}[System of fields functor]
 \label{def:colim-fields}
-If $\cC$ is a disk-like $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
+If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.
 That is, for each decomposition $x$ there is a map
 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps
 above, and $\cl{\cC}(W)$ is universal with respect to these properties.
 \end{defn}
 
 \begin{defn}[System of fields functor, $A_\infty$ case]
-When $\cC$ is an $A_\infty$ disk-like $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ 
+When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ 
 is defined as above, as the colimit of $\psi_{\cC;W}$.
 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.
 \end{defn}
@@ -1585,16 +1585,16 @@
 
 \subsection{Modules}
 
-Next we define ordinary and $A_\infty$ disk-like $n$-category modules.
-The definition will be very similar to that of disk-like $n$-categories,
+Next we define ordinary and $A_\infty$ $n$-category modules.
+The definition will be very similar to that of $n$-categories,
 but with $k$-balls replaced by {\it marked $k$-balls,} defined below.
 
 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
 in the context of an $m{+}1$-dimensional TQFT.
-Such a $W$ gives rise to a module for the disk-like $n$-category associated to $\bd W$.
+Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
 This will be explained in more detail as we present the axioms.
 
-Throughout, we fix a disk-like $n$-category $\cC$.
+Throughout, we fix an $n$-category $\cC$.
 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category.
 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases.
 
@@ -1650,7 +1650,7 @@
 
 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
 
-If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
+If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
 
 \begin{lem}[Boundary from domain and range]
@@ -1857,7 +1857,7 @@
 \end{enumerate}
 \end{module-axiom}
 
-As in the disk-like $n$-category definition, once we have product morphisms we can define
+As in the $n$-category definition, once we have product morphisms we can define
 collar maps $\cM(M)\to \cM(M)$.
 Note that there are two cases:
 the collar could intersect the marking of the marked ball $M$, in which case
@@ -1870,7 +1870,7 @@
 \medskip
 
 There are two alternatives for the next axiom, according whether we are defining
-modules for ordinary or $A_\infty$ disk-like $n$-categories.
+modules for ordinary $n$-categories or $A_\infty$ $n$-categories.
 In the ordinary case we require
 
 \begin{module-axiom}[\textup{\textbf{[ordinary version]}} Extended isotopy invariance in dimension $n$]
@@ -1903,14 +1903,14 @@
 
 \medskip
 
-Note that the above axioms imply that a disk-like $n$-category module has the structure
-of a disk-like $n{-}1$-category.
+Note that the above axioms imply that an $n$-category module has the structure
+of an $n{-}1$-category.
 More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$,
 where $X$ is a $k$-ball and in the product $X\times J$ we pinch 
 above the non-marked boundary component of $J$.
 (More specifically, we collapse $X\times P$ to a single point, where
 $P$ is the non-marked boundary component of $J$.)
-Then $\cE$ has the structure of a disk-like $n{-}1$-category.
+Then $\cE$ has the structure of an $n{-}1$-category.
 
 All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds
 are oriented or Spin (but not unoriented or $\text{Pin}_\pm$).
@@ -1922,12 +1922,12 @@
 
 \medskip
 
-We now give some examples of modules over ordinary and $A_\infty$ disk-like $n$-categories.
+We now give some examples of modules over ordinary and $A_\infty$ $n$-categories.
 
 \begin{example}[Examples from TQFTs]
 \rm
 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold,
-and $\cF(W)$ the disk-like $j$-category associated to $W$.
+and $\cF(W)$ the $j$-category associated to $W$.
 Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$.
 Define a $\cF(W)$ module $\cF(Y)$ as follows.
 If $M = (B, N)$ is a marked $k$-ball with $k<j$ let 
@@ -1940,7 +1940,7 @@
 \rm
 In the previous example, we can instead define
 $\cF(Y)(M)\deq \bc_*((B\times W) \cup (N\times Y), c; \cF)$ (when $\dim(M) = n$)
-and get a module for the $A_\infty$ disk-like $n$-category associated to $\cF$ as in 
+and get a module for the $A_\infty$ $n$-category associated to $\cF$ as in 
 Example \ref{ex:blob-complexes-of-balls}.
 \end{example}
 
@@ -1965,7 +1965,7 @@
 \subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}
 
-Fix an ordinary or $A_\infty$ disk-like $n$-category  $\cC$.
+Fix an ordinary $n$-category or $A_\infty$ $n$-category  $\cC$.
 Let $W$ be a $k$-manifold ($k\le n$),
 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
@@ -2021,19 +2021,19 @@
 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold 
 $D\times Y_i \sub \bd(D\times W)$.
 It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$
-has the structure of a disk-like $n{-}k$-category.
+has the structure of an $n{-}k$-category.
 
 \medskip
 
 We will use a simple special case of the above 
 construction to define tensor products 
 of modules.
-Let $\cM_1$ and $\cM_2$ be modules for a disk-like $n$-category $\cC$.
+Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$.
 (If $k=1$ and our manifolds are oriented, then one should be 
 a left module and the other a right module.)
 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$.
 Define the tensor product $\cM_1 \tensor \cM_2$ to be the 
-disk-like $n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
+$n{-}1$-category associated as above to $J$ with its boundary labeled by $\cM_1$ and $\cM_2$.
 This of course depends (functorially)
 on the choice of 1-ball $J$.
 
@@ -2738,19 +2738,19 @@
 
 \medskip
 
-We end this subsection with some remarks about Morita equivalence of disk-like $n$-categories.
+We end this subsection with some remarks about Morita equivalence of disklike $n$-categories.
 Recall that two 1-categories $\cC$ and $\cD$ are Morita equivalent if and only if they are equivalent
 objects in the 2-category of (linear) 1-categories, bimodules, and intertwiners.
-Similarly, we define two disk-like $n$-categories to be Morita equivalent if they are equivalent objects in the
+Similarly, we define two disklike $n$-categories to be Morita equivalent if they are equivalent objects in the
 $n{+}1$-category of sphere modules.
 
-Because of the strong duality enjoyed by disk-like $n$-categories, the data for such an equivalence lives only in 
+Because of the strong duality enjoyed by disklike $n$-categories, the data for such an equivalence lives only in 
 dimensions 1 and $n+1$ (the middle dimensions come along for free).
 The $n{+}1$-dimensional part of the data must be invertible and satisfy
 identities corresponding to Morse cancellations in $n$-manifolds.
 We will treat this in detail for the $n=2$ case; the case for general $n$ is very similar.
 
-Let $\cC$ and $\cD$ be (unoriented) disk-like 2-categories.
+Let $\cC$ and $\cD$ be (unoriented) disklike 2-categories.
 Let $\cS$ denote the 3-category of 2-category sphere modules.
 The 1-dimensional part of the data for a Morita equivalence between $\cC$ and $\cD$ is a 0-sphere module $\cM = {}_\cC\cM_\cD$ 
 (categorified bimodule) connecting $\cC$ and $\cD$.
diff -r 7abe7642265e -r a0fd6e620926 text/tqftreview.tex
--- a/text/tqftreview.tex	Tue Aug 09 23:55:13 2011 -0700
+++ b/text/tqftreview.tex	Sun Sep 25 14:33:30 2011 -0600
@@ -19,7 +19,7 @@
 We do this in detail for $n=1,2$ (\S\ref{sec:example:traditional-n-categories(fields)}) 
 and more informally for general $n$.
 In the other direction, 
-our preferred definition of a disk-like $n$-category in \S\ref{sec:ncats} is essentially
+our preferred definition of an $n$-category in \S\ref{sec:ncats} is essentially
 just a system of fields restricted to balls of dimensions 0 through $n$;
 one could call this the ``local" part of a system of fields.