--- a/.hgignore Wed May 12 15:57:20 2010 -0700
+++ b/.hgignore Wed May 12 18:26:20 2010 -0500
@@ -5,6 +5,7 @@
*.aux
*.log
*.out
+*.toc
*.synctex.gz
# the article PDF
--- a/blob1.tex Wed May 12 15:57:20 2010 -0700
+++ b/blob1.tex Wed May 12 18:26:20 2010 -0500
@@ -1,4 +1,4 @@
-\documentclass[11pt,leqno]{amsart}
+\documentclass[11pt,leqno]{article}
%\usepackage{amsthm}
@@ -45,7 +45,8 @@
{\bf then don't read this version,} as a more complete version will be available in a couple of months.
\nn{maybe to do: add appendix on various versions of acyclic models}
-%\tableofcontents
+
+\tableofcontents
--- a/preamble.tex Wed May 12 15:57:20 2010 -0700
+++ b/preamble.tex Wed May 12 18:26:20 2010 -0500
@@ -2,7 +2,7 @@
%!TEX root = blob1.tex
%this ensures the arxiv doesn't try to start TeXing here.
-\usepackage{amsmath,amssymb,amsfonts}
+\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{ifpdf}
%\ifpdf
@@ -11,6 +11,7 @@
\usepackage[all,color]{xy}
%\fi
+\usepackage[section]{placeins}
\usepackage{leftidx}
\SelectTips{cm}{}
--- a/text/ncat.tex Wed May 12 15:57:20 2010 -0700
+++ b/text/ncat.tex Wed May 12 18:26:20 2010 -0500
@@ -23,21 +23,16 @@
\medskip
-Consider first ordinary $n$-categories.
-\nn{Actually, we're doing both plain and infinity cases here}
-We need a set (or sets) of $k$-morphisms for each $0\le k \le n$.
-We must decide on the ``shape" of the $k$-morphisms.
-Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...).
+There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical chape.
+Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on).
Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$,
a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,
and so on.
(This allows for strict associativity.)
-Still other definitions \nn{need refs for all these; maybe the Leinster book \cite{MR2094071}}
+Still other definitions (see, for example, \cite{MR2094071})
model the $k$-morphisms on more complicated combinatorial polyhedra.
-We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to
-the standard $k$-ball.
-In other words,
+For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball:
\begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms}
For any $k$-manifold $X$ homeomorphic
@@ -45,11 +40,10 @@
$\cC_k(X)$.
\end{preliminary-axiom}
-Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
+By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
We {\it do not} assume that it is equipped with a
-preferred homeomorphism to the standard $k$-ball.
-The same goes for ``a $k$-sphere" below.
+preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on
@@ -84,21 +78,21 @@
Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
(Actually, this is only true in the oriented case, with 1-morphsims parameterized
by oriented 1-balls.)
-For $k>1$ and in the presence of strong duality the domain/range division makes less sense.
-\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah}
-We prefer to combine the domain and range into a single entity which we call the
+For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{A \tensor B^*}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place.
+
+Instead, we combine the domain and range into a single entity which we call the
boundary of a morphism.
Morphisms are modeled on balls, so their boundaries are modeled on spheres:
-\nn{perhaps it's better to define $\cC(S^k)$ as a colimit, rather than making it new data}
-
\begin{axiom}[Boundaries (spheres)]
For each $0 \le k \le n-1$, we have a functor $\cC_k$ from
the category of $k$-spheres and
homeomorphisms to the category of sets and bijections.
\end{axiom}
-(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.)
+In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries, and again often omit the subscript.
+
+In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
\begin{axiom}[Boundaries (maps)]\label{nca-boundary}
For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$.
@@ -108,7 +102,7 @@
(Note that the first ``$\bd$" above is part of the data for the category,
while the second is the ordinary boundary of manifolds.)
-Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$.
+Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$.
Most of the examples of $n$-categories we are interested in are enriched in the following sense.
The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
--- a/text/top_matter.tex Wed May 12 15:57:20 2010 -0700
+++ b/text/top_matter.tex Wed May 12 18:26:20 2010 -0500
@@ -2,15 +2,11 @@
\title{Blob Homology}
-\author{Scott~Morrison}
-\address{
-}%
-\email{scott@tqft.net} \urladdr{http://tqft.net/}
+\author{Scott~Morrison and Kevin~Walker}
+%\email{scott@tqft.net} \urladdr{http://tqft.net/}
-\author{Kevin~Walker}
-\address{
-}%
-\email{kevin@canyon23.net} \urladdr{http://canyon23.net/math/}
+%\author{Kevin~Walker}
+%\email{kevin@canyon23.net} \urladdr{http://canyon23.net/math/}
\date{