# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1224993475 0 # Node ID 0ffcbbd8019c13fa118349610dffb7c9dafb9054 # Parent 0047a1211c3bdddbbe65f7f90a0c896bafa6a5e3 minor cleanup of the start of the hochschild section 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