# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1289179390 -32400
# Node ID e1840aaa31ffdc019adb0c87e0b974c1c3df50cc
# Parent  294c6b2ab72323709b7370404edf1e0ed9609de0# Parent  1b41a54d8d18134864e8c77f81f00b9327be29a2
Automated merge with https://tqft.net/hg/blob/

diff -r 1b41a54d8d18 -r e1840aaa31ff pnas/pnas.tex
--- a/pnas/pnas.tex	Mon Nov 08 10:22:02 2010 +0900
+++ b/pnas/pnas.tex	Mon Nov 08 10:23:10 2010 +0900
@@ -223,6 +223,10 @@
 homeomorphisms to the category of sets and bijections.
 \end{axiom}
 
+Note that the functoriality in the above axiom allows us to operate via
+homeomorphisms which are not the identity on the boundary of the $k$-ball.
+The action of these homeomorphisms gives the ``strong duality" structure.
+
 Next we consider domains and ranges of $k$-morphisms.
 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism
 into domain and range --- the duality operations can convert domain to range and vice-versa.
@@ -235,6 +239,15 @@
 These maps, for various $X$, comprise a natural transformation of functors.
 \end{axiom}
 
+For $c\in \cl{\cC}_{k-1}(\bd X)$ we let $\cC_k(X; c)$ denote the preimage $\bd^{-1}(c)$.
+
+Many of the examples we are interested in are enriched in some auxiliary category $\cS$
+(e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces).
+This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure
+of an object of $\cS$, and all of the structure maps of the category (above and below) are
+compatible with the $\cS$ structure on $\cC_n(X; c)$.
+
+
 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere.
 
 \begin{lem}
@@ -369,6 +382,8 @@
 Maybe just a single remark that we are omitting some details which appear in our
 longer paper.}
 \nn{SM: for now I disagree: the space expense is pretty minor, and it always us to be "in principle" complete. Let's see how we go for length.}
+\nn{KW: It's not the length I'm worried about --- I was worried about distracting the reader
+with an arcane technical issue.  But we can decide later.}
 
 A \emph{ball decomposition} of $W$ is a 
 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
@@ -493,14 +508,13 @@
 
 \begin{property}[Contractibility]
 \label{property:contractibility}%
-With field coefficients, the blob complex on an $n$-ball is contractible in the sense 
-that it is homotopic to its $0$-th homology.
-Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces 
-associated by the system of fields $\cF$ to balls.
+The blob complex on an $n$-ball is contractible in the sense 
+that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category.
 \begin{equation*}
-\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)}
+\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
 \end{equation*}
 \end{property}
+\nn{maybe should say something about the $A_\infty$ case}
 
 \begin{proof}(Sketch)
 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram
@@ -509,7 +523,6 @@
 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
 \end{proof}
 
-
 \subsection{Specializations}
 \label{sec:specializations}
 
@@ -517,13 +530,15 @@
 
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
+\nn{Plain n-categories only?}
 The $0$-th blob homology of $X$ is the usual 
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
-by $\cF$.
+by $\cC$.
 \begin{equation*}
-H_0(\bc_*(X;\cF)) \iso A_{\cF}(X)
+H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
 \end{equation*}
 \end{thm}
+This follows from the fact that the $0$-th homology of a homotopy colimit of \todo{something plain} is the usual colimit, or directly from the explicit description of the blob complex.
 
 \begin{thm}[Hochschild homology when $X=S^1$]
 \label{thm:hochschild}
diff -r 1b41a54d8d18 -r e1840aaa31ff text/intro.tex
--- a/text/intro.tex	Mon Nov 08 10:22:02 2010 +0900
+++ b/text/intro.tex	Mon Nov 08 10:23:10 2010 +0900
@@ -277,7 +277,7 @@
 \end{equation*}
 \end{property}
 
-Properties \ref{property:functoriality} will be immediate from the definition given in
+Property \ref{property:functoriality} will be immediate from the definition given in
 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and 
 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.