# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1277744593 25200
# Node ID 291f82fb79b59341f116aa705794606714544adf
# Parent  37f036dda03c19db5c442f48134f2b92ad004dbb
mostly hochschild stuff

diff -r 37f036dda03c -r 291f82fb79b5 text/basic_properties.tex
--- a/text/basic_properties.tex	Mon Jun 28 08:54:36 2010 -0700
+++ b/text/basic_properties.tex	Mon Jun 28 10:03:13 2010 -0700
@@ -95,19 +95,19 @@
 For the next proposition we will temporarily restore $n$-manifold boundary
 conditions to the notation.
 
-Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$.
+Let $X$ be an $n$-manifold, $\bd X = Y \cup Y \cup Z$.
 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$
 with boundary $Z\sgl$.
-Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
+Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$,
 we have the blob complex $\bc_*(X; a, b, c)$.
-If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
+If $b = a$, then we can glue up blob diagrams on
 $X$ to get blob diagrams on $X\sgl$.
 This proves Property \ref{property:gluing-map}, which we restate here in more detail.
 
 \textbf{Property \ref{property:gluing-map}.}\emph{
 There is a natural chain map
 \eq{
-    \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
+    \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl).
 }
 The sum is over all fields $a$ on $Y$ compatible at their
 ($n{-}2$-dimensional) boundaries with $c$.
diff -r 37f036dda03c -r 291f82fb79b5 text/hochschild.tex
--- a/text/hochschild.tex	Mon Jun 28 08:54:36 2010 -0700
+++ b/text/hochschild.tex	Mon Jun 28 10:03:13 2010 -0700
@@ -19,7 +19,7 @@
 to find a more ``local" description of the Hochschild complex.
 
 Let $C$ be a *-1-category.
-Then specializing the definitions from above to the case $n=1$ we have: \nn{mention that this is dual to the way we think later} \nn{mention that this has the nice side effect of making everything splittable away from the marked points}
+Then specializing the definitions from above to the case $n=1$ we have:
 \begin{itemize}
 \item $\cC(pt) = \ob(C)$ .
 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
@@ -31,7 +31,7 @@
 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by
 composing the morphism labels of the points.
 Note that we also need the * of *-1-category here in order to make all the morphisms point
-the same way. \nn{Wouldn't it be better to just do the oriented version here? -S}
+the same way.
 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single
 point (at some standard location) labeled by $x$.
 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
@@ -204,7 +204,8 @@
 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$.
 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either
 (a) the point * is not on the boundary of any blob or
-(b) there are no labeled points or blob boundaries within distance $\ep$ of *.
+(b) there are no labeled points or blob boundaries within distance $\ep$ of *,
+other than blob boundaries at * itself.
 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small.
 Let $b$ be a blob diagram in $F_*^\ep$.
 Define $f(b)$ to be the result of moving any blob boundary points which lie on *
@@ -236,7 +237,9 @@
 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding 
 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction
 of $x$ to $N_\ep$.
-If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, \nn{I don't think we need to consider sums here}
+If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, 
+\nn{SM: I don't think we need to consider sums here}
+\nn{KW: It depends on whether we allow linear combinations of fields outside of twig blobs}
 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let
 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$,
 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$.
diff -r 37f036dda03c -r 291f82fb79b5 text/tqftreview.tex
--- a/text/tqftreview.tex	Mon Jun 28 08:54:36 2010 -0700
+++ b/text/tqftreview.tex	Mon Jun 28 10:03:13 2010 -0700
@@ -209,6 +209,15 @@
 with codimension $i$ cells labeled by $i$-morphisms of $C$.
 We'll spell this out for $n=1,2$ and then describe the general case.
 
+This way of decorating an $n$-manifold with an $n$-category is sometimes referred to
+as a ``string diagram".
+It can be thought of as (geometrically) dual to a pasting diagram.
+One of the advantages of string diagrams over pasting diagrams is that one has more
+flexibility in slicing them up in various ways.
+In addition, string diagrams are traditional in quantum topology.
+The diagrams predate by many years the terms ``string diagram" and ``quantum topology".
+\nn{?? cite penrose, kauffman, jones(?)}
+
 If $X$ has boundary, we require that the cell decompositions are in general
 position with respect to the boundary --- the boundary intersects each cell
 transversely, so cells meeting the boundary are mere half-cells.