
diff -r 853376c08d76 -r ef36cdefb130 text/hochschild.tex
--- a/text/hochschild.tex	Sun Jun 27 12:28:06 2010 -0700
+++ b/text/hochschild.tex	Sun Jun 27 12:53:11 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:
+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}
 \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.
+the same way. \nn{Wouldn't it be better to just do the oriented version here? -S}
 \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
@@ -130,7 +130,7 @@
 \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j).
 \end{align*}
 The cone of each chain map is acyclic.
-In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact.
+In the first case, this is because the `rows' indexed by $i$ are acyclic since $\cP_i$ is exact.
 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free.
 Because the cones are acyclic, the chain maps are quasi-isomorphisms.
 Composing one with the inverse of the other, we obtain the desired quasi-isomorphism
@@ -236,7 +236,7 @@
 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 $u=\sum z_i$,
+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}
 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 $y_i - s(y_i)$.




