
diff -r 0047a1211c3b -r 0ffcbbd8019c text/hochschild.tex
--- a/text/hochschild.tex	Wed Oct 22 21:56:42 2008 +0000
+++ b/text/hochschild.tex	Sun Oct 26 03:57:55 2008 +0000
@@ -1,33 +1,8 @@
-In this section we analyze the blob complex in dimension $n=1$
-and find that for $S^1$ the homology of the blob complex is the
-Hochschild homology of the category (algebroid) that we started with.
-\nn{or maybe say here that the complexes are quasi-isomorphic?  in general,
-should perhaps put more emphasis on the complexes and less on the homology.}
-
-Notation: $HB_i(X) = H_i(\bc_*(X))$.
+%!TEX root = ../blob1.tex
 
-Let us first note that there is no loss of generality in assuming that our system of
-fields comes from a category.
-(Or maybe (???) there {\it is} a loss of generality.
-Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
-thought of as the morphisms of a 1-category $C$.
-More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$
-are $A(I; a, b)$, and composition is given by gluing.
-If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change
-and neither does $A(I; a, b) = HB_0(I; a, b)$.
-But what about $HB_i(I; a, b)$ for $i > 0$?
-Might these higher blob homology groups be different?
-Seems unlikely, but I don't feel like trying to prove it at the moment.
-In any case, we'll concentrate on the case of fields based on 1-category
-pictures for the rest of this section.)
-
-(Another question: $\bc_*(I)$ is an $A_\infty$-category.
-How general of an $A_\infty$-category is it?
-Given an arbitrary $A_\infty$-category can one find fields and local relations so
-that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category?
-Probably not, unless we generalize to the case where $n$-morphisms are complexes.)
-
-Continuing...
+In this section we analyze the blob complex in dimension $n=1$
+and find that for $S^1$ the blob complex is homotopy equivalent to the 
+Hochschild complex of the category (algebroid) that we started with.
 
 Let $C$ be a *-1-category.
 Then specializing the definitions from above to the case $n=1$ we have:
@@ -50,26 +25,13 @@
 Thus we can, if we choose, restrict the blob twig labels to things of this form.
 \end{itemize}
 
-We want to show that $HB_*(S^1)$ is naturally isomorphic to the
-Hochschild homology of $C$.
-\nn{Or better that the complexes are homotopic
-or quasi-isomorphic.}
+We want to show that $\bc_*(S^1)$ is homotopy equivalent to the
+Hochschild complex of $C$.
+(Note that both complexes are free (and hence projective), so it suffices to show that they
+are quasi-isomorphic.)
 In order to prove this we will need to extend the blob complex to allow points to also
 be labeled by elements of $C$-$C$-bimodules.
-%Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product
-%(over $C$) of $C$-$C$-bimodules.
-%Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps.
-%Now we can define the blob complex for $S^1$.
-%This complex is the sum of complexes with a fixed cyclic tuple of bimodules present.
-%If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding
-%to the cyclic 1-tuple $(M)$.
-%In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled
-%by an element of $M$ and the remaining points are labeled by morphisms of $C$.
-%It's clear that $G_*(C)$ is isomorphic to the original bimodule-less
-%blob complex for $S^1$.
-%\nn{Is it really so clear?  Should say more.}
 
-%\nn{alternative to the above paragraph:}
 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$.
 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling
@@ -80,17 +42,6 @@
 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$
 and elements of $C$ at variable other points.
 
-\todo{Some orphaned questions:}
-\nn{Or maybe we should claim that $M \to K_*(M)$ is the/a derived coend.
-Or maybe that $K_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild
-complex of $M$.}
-
-\nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex?
-Do we need a map from hoch to blob?
-Does the above exactness and contractibility guarantee such a map without writing it
-down explicitly?
-Probably it's worth writing down an explicit map even if we don't need to.}
-
 
 We claim that
 \begin{thm}
@@ -98,11 +49,6 @@
 usual Hochschild complex for $C$.
 \end{thm}
 
-\nn{Note that since both complexes are free (in particular, projective),
-quasi-isomorphic implies homotopy equivalent.  
-This applies to the two claims below also.
-Thanks to Peter Teichner for pointing this out to me.}
-
 This follows from two results. First, we see that
 \begin{lem}
 \label{lem:module-blob}%
@@ -211,7 +157,7 @@
 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$.
 $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point *
 is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be.
-In other words, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$.
+In particular, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$.
 
 We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows.
 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if
@@ -225,7 +171,6 @@
 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
 in a neighborhood $B_\ep$ of $*$, except perhaps $*$, and $B_\ep$ is either disjoint from or contained in every blob.
 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$.
-\nn{rest of argument goes similarly to above}
 
 We define a degree $1$ chain map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram.
 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $B_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction




