diff -r 15b13864b02e -r 7340ab80db25 text/hochschild.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/text/hochschild.tex	Sun Jun 08 21:34:46 2008 +0000
@@ -0,0 +1,346 @@
+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: $HB_i(X) = H_i(\bc_*(X))$.
+
+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...
+
+Let $C$ be a *-1-category.
+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 $c \in \cC(\bd R)$.
+Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
+points in the interior
+of $R$, each labeled by a morphism of $C$.
+The intervals between the points are labeled by objects of $C$, consistent with
+the boundary condition $c$ and the domains and ranges of the point labels.
+\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.
+\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
+form $y - \chi(e(y))$.
+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.}
+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 $F_*(S^1, (p_i), (M_i))$.
+The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling
+other points.
+The blob twig labels lie in kernels of evaluation maps.
+(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.)
+Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point.
+In other words, fields for $F_*(M)$ 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 $M \to F_*(M)$ is the/a derived coend.
+Or maybe that $F_*(M)$ 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}
+The blob complex $\bc_*(S^1; C)$ on the circle is quasi-isomorphic to the
+usual Hochschild complex for $C$.
+\end{thm}
+
+This follows from two results. First, we see that
+\begin{lem}
+\label{lem:module-blob}%
+The complex $F_*(C)$ (here $C$ is being thought of as a
+$C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex
+$\bc_*(S^1; C)$. (Proof later.)
+\end{lem}
+
+Next, we show that for any $C$-$C$-bimodule $M$,
+\begin{prop}
+The complex $F_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual
+Hochschild complex of $M$.
+\end{prop}
+\begin{proof}
+First, since we're working over $\Complex$, note that saying two complexes are quasi-isomorphic simply means they have isomorphic homologies.
+\todo{We really don't want to work over $\Complex$, though; it would be nice to talk about torsion!}
+
+Recall that the usual Hochschild complex of $M$ is uniquely determined, up to quasi-isomorphism, by the following properties:
+\begin{enumerate}
+\item \label{item:hochschild-additive}%
+$HC_*(M_1 \oplus M_2) \cong HC_*(M_1) \oplus HC_*(M_2)$.
+\item \label{item:hochschild-exact}%
+An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an
+exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$.
+\item \label{item:hochschild-free}%
+$HC_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is
+quasi-isomorphic to the 0-step complex $C$.
+\item \label{item:hochschild-coinvariants}%
+$HH_0(M)$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$.
+\end{enumerate}
+(Together, these just say that Hochschild homology is `the derived functor of coinvariants'.)
+We'll first explain why these properties are characteristic. Take some
+$C$-$C$ bimodule $M$. If $M$ is free, that is, a direct sum of copies of
+$C \tensor C$, then properties \ref{item:hochschild-additive} and
+\ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some
+free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we
+have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a
+short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M)
+\to 0$. Such a sequence gives a long exact sequence on homology
+\begin{equation*}
+%\begin{split}
+\cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\
+%\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M).
+%\end{split}
+\end{equation*}
+For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties
+\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so
+$HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}.
+
+This tells us how to
+compute every homology group of $HC_*(M)$; we already know $HH_0(M)$
+(it's just coinvariants, by property \ref{item:hochschild-coinvariants}),
+and higher homology groups are determined by lower ones in $HC_*(K)$, and
+hence recursively as coinvariants of some other bimodule.
+
+The proposition then follows from the following lemmas, establishing that $F_*$ has precisely these required properties.
+\begin{lem}
+\label{lem:hochschild-additive}%
+Directly from the definition, $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$.
+\end{lem}
+\begin{lem}
+\label{lem:hochschild-exact}%
+An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an
+exact sequence $0 \to F_*(M_1) \into F_*(M_2) \onto F_*(M_3) \to 0$.
+\end{lem}
+\begin{lem}
+\label{lem:hochschild-free}%
+$F_*(C\otimes C)$ is quasi-isomorphic to the 0-step complex $C$.
+\end{lem}
+\begin{lem}
+\label{lem:hochschild-coinvariants}%
+$H_0(F_*(M))$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$.
+\end{lem}
+
+The remainder of this section is devoted to proving Lemmas
+\ref{lem:module-blob},
+\ref{lem:hochschild-exact}, \ref{lem:hochschild-free} and
+\ref{lem:hochschild-coinvariants}.
+\end{proof}
+
+\begin{proof}[Proof of Lemma \ref{lem:module-blob}]
+We show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$.
+$F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point *
+is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be.
+In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$.
+
+We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows.
+If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if
+* is a labeled point in $y$.
+Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.
+Let $x \in \bc_*(S^1)$.
+Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
+$x$ with $y$.
+It is easy to check that $s$ is a chain map and $s \circ i = \id$.
+
+Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
+in a neighborhood $B_\ep$ of *, except perhaps *.
+Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$.
+\nn{rest of argument goes similarly to above}
+\end{proof}
+\begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}]
+\todo{}
+\end{proof}
+\begin{proof}[Proof of Lemma \ref{lem:hochschild-free}]
+We show that $F_*(C\otimes C)$ is
+quasi-isomorphic to the 0-step complex $C$.
+
+Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
+the point $*$ is $1 \otimes 1 \in C\otimes C$.
+We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
+
+Fix a small $\ep > 0$.
+Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
+Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
+generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from
+or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$.
+For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
+labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
+(See Figure xxxx.)
+Note that $y - s_\ep(y) \in U(B_\ep)$.
+\nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.}
+
+Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows.
+Let $x \in F^\ep_*$ be a blob diagram.
+If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
+$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
+If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
+Let $y_i$ be the restriction of $z_i$ to $B_\ep$.
+Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
+and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
+Define $j_\ep(x) = \sum x_i$.
+\nn{need to check signs coming from blob complex differential}
+
+Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.
+
+The key property of $j_\ep$ is
+\eq{
+    \bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
+}
+where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field
+mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.
+Note that $\sigma_\ep(x) \in F'_*$.
+
+If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$
+is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
+One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
+$\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
+Instead, we'll be less ambitious and just show that
+$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
+
+If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have
+$x \in F_*^\ep$.
+(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
+finitely many blob diagrams.)
+Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map
+$F'_* \sub F_*(C\otimes C)$ is surjective on homology.
+If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
+and
+\eq{
+    \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
+}
+Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
+This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
+
+Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob.
+We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence.
+
+First, a lemma:  Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with
+$S^1$ replaced some (any) neighborhood of $* \in S^1$.
+Then $G''_*$ and $G'_*$ are both contractible
+and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.
+For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
+$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
+For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe
+in ``basic properties" section above} away from $*$.
+Thus any cycle lies in the image of the normal blob complex of a disjoint union
+of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}).
+Actually, we need the further (easy) result that the inclusion
+$G''_* \to G'_*$ induces an isomorphism on $H_0$.
+
+Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that
+for all $x \in F'_*$ we have
+\eq{
+    x - \bd h(x) - h(\bd x) \in F''_* .
+}
+Since $F'_0 = F''_0$, we can take $h_0 = 0$.
+Let $x \in F'_1$, with single blob $B \sub S^1$.
+If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$.
+If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$).
+Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$.
+Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$.
+Define $h_1(x) = y$.
+The general case is similar, except that we have to take lower order homotopies into account.
+Let $x \in F'_k$.
+If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$.
+Otherwise, let $B$ be the outermost blob of $x$ containing $*$.
+By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$.
+So $x' \in G'_l$ for some $l \le k$.
+Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$.
+Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$.
+Define $h_k(x) = y \bullet p$.
+This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence.
+\nn{need to say above more clearly and settle on notation/terminology}
+
+Finally, we show that $F''_*$ is contractible.
+\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
+Let $x$ be a cycle in $F''_*$.
+The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
+ball $B \subset S^1$ containing the union of the supports and not containing $*$.
+Adding $B$ as a blob to $x$ gives a contraction.
+\nn{need to say something else in degree zero}
+\end{proof}
+\begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}]
+\todo{}
+\end{proof}
+
+We can also describe explicitly a map from the standard Hochschild
+complex to the blob complex on the circle. \nn{What properties does this
+map have?}
+
+\begin{figure}%
+$$\mathfig{0.6}{barycentric/barycentric}$$
+\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to
+the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.}
+\label{fig:Hochschild-example}%
+\end{figure}
+
+As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly.
+The edges marked $x, y$ and $z$ carry the $1$-chains
+\begin{align*}
+x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\
+y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\
+z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab}
+\end{align*}
+and the $2$-chain labelled $A$ is
+\begin{equation*}
+A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}.
+\end{equation*}
+Note that we then have
+\begin{equation*}
+\bdy A = x+y+z.
+\end{equation*}
+
+In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations,
+$$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$
+with ... (hmmm, problems making this precise; you need to decide where to put the labels, but then it's hard to make an honest chain map!)