580 

   580 \section{Hochschild homology when $n=1$}

   581 

   581 \label{sec:hochschild}

   582 

   582 \input{text/hochschild}

   583 \section{$n=1$ and Hochschild homology}

   584 

   585 In this section we analyze the blob complex in dimension $n=1$

   586 and find that for $S^1$ the homology of the blob complex is the

   587 Hochschild homology of the category (algebroid) that we started with.

   588 \nn{or maybe say here that the complexes are quasi-isomorphic?  in general,

   589 should perhaps put more emphasis on the complexes and less on the homology.}

   590 

   591 Notation: $HB_i(X) = H_i(\bc_*(X))$.

   592 

   593 Let us first note that there is no loss of generality in assuming that our system of

   594 fields comes from a category.

   595 (Or maybe (???) there {\it is} a loss of generality.

   596 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be

   597 thought of as the morphisms of a 1-category $C$.

   598 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$

   599 are $A(I; a, b)$, and composition is given by gluing.

   600 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change

   601 and neither does $A(I; a, b) = HB_0(I; a, b)$.

   602 But what about $HB_i(I; a, b)$ for $i > 0$?

   603 Might these higher blob homology groups be different?

   604 Seems unlikely, but I don't feel like trying to prove it at the moment.

   605 In any case, we'll concentrate on the case of fields based on 1-category

   606 pictures for the rest of this section.)

   607 

   608 (Another question: $\bc_*(I)$ is an $A_\infty$-category.

   609 How general of an $A_\infty$-category is it?

   610 Given an arbitrary $A_\infty$-category can one find fields and local relations so

   611 that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category?

   612 Probably not, unless we generalize to the case where $n$-morphisms are complexes.)

   613 

   614 Continuing...

   615 

   616 Let $C$ be a *-1-category.

   617 Then specializing the definitions from above to the case $n=1$ we have:

   618 \begin{itemize}

   619 \item $\cC(pt) = \ob(C)$ .

   620 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.

   621 Then an element of $\cC(R; c)$ is a collection of (transversely oriented)

   622 points in the interior

   623 of $R$, each labeled by a morphism of $C$.

   624 The intervals between the points are labeled by objects of $C$, consistent with

   625 the boundary condition $c$ and the domains and ranges of the point labels.

   626 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by

   627 composing the morphism labels of the points.

   628 Note that we also need the * of *-1-category here in order to make all the morphisms point

   629 the same way.

   630 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single

   631 point (at some standard location) labeled by $x$.

   632 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the

   633 form $y - \chi(e(y))$.

   634 Thus we can, if we choose, restrict the blob twig labels to things of this form.

   635 \end{itemize}

   636 

   637 We want to show that $HB_*(S^1)$ is naturally isomorphic to the

   638 Hochschild homology of $C$.

   639 \nn{Or better that the complexes are homotopic

   640 or quasi-isomorphic.}

   641 In order to prove this we will need to extend the blob complex to allow points to also

   642 be labeled by elements of $C$-$C$-bimodules.

   643 %Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product

   644 %(over $C$) of $C$-$C$-bimodules.

   645 %Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps.

   646 %Now we can define the blob complex for $S^1$.

   647 %This complex is the sum of complexes with a fixed cyclic tuple of bimodules present.

   648 %If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding

   649 %to the cyclic 1-tuple $(M)$.

   650 %In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled

   651 %by an element of $M$ and the remaining points are labeled by morphisms of $C$.

   652 %It's clear that $G_*(C)$ is isomorphic to the original bimodule-less

   653 %blob complex for $S^1$.

   654 %\nn{Is it really so clear?  Should say more.}

   655 

   656 %\nn{alternative to the above paragraph:}

   657 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.

   658 We define a blob-like complex $F_*(S^1, (p_i), (M_i))$.

   659 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling

   660 other points.

   661 The blob twig labels lie in kernels of evaluation maps.

   662 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.)

   663 Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point.

   664 In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$

   665 and elements of $C$ at variable other points.

   666 

   667 We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild

   668 homology of $M$.

   669 \nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend.

   670 Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild

   671 complex of $M$.}

   672 This follows from the following lemmas:

   673 \begin{itemize}

   674 \item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$.

   675 \item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$

   676 gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$.

   677 (See below for proof.)

   678 \item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is

   679 quasi-isomorphic to the 0-step complex $C$.

   680 (See below for proof.)

   681 \item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is quasi-isomorphic to $\bc_*(S^1)$.

   682 (See below for proof.)

   683 \end{itemize}

   684 

   685 First we show that $F_*(C\otimes C)$ is

   686 quasi-isomorphic to the 0-step complex $C$.

   687 

   688 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of

   689 the point $*$ is $1 \otimes 1 \in C\otimes C$.

   690 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.

   691 

   692 Fix a small $\ep > 0$.

   693 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.

   694 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex

   695 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from

   696 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$.

   697 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$

   698 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.

   699 (See Figure xxxx.)

   700 Note that $y - s_\ep(y) \in U(B_\ep)$.

   701 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.}

   702 

   703 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows.

   704 Let $x \in F^\ep_*$ be a blob diagram.

   705 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to

   706 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.

   707 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.

   708 Let $y_i$ be the restriction of $z_i$ to $B_\ep$.

   709 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,

   710 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.

   711 Define $j_\ep(x) = \sum x_i$.

   712 \nn{need to check signs coming from blob complex differential}

   713 

   714 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.

   715 

   716 The key property of $j_\ep$ is

   717 \eq{

   718     \bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,

   719 }

   720 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field

   721 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.

   722 Note that $\sigma_\ep(x) \in F'_*$.

   723 

   724 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$

   725 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.

   726 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller

   727 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.

   728 Instead, we'll be less ambitious and just show that

   729 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.

   730 

   731 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have

   732 $x \in F_*^\ep$.

   733 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of

   734 finitely many blob diagrams.)

   735 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map

   736 $F'_* \sub F_*(C\otimes C)$ is surjective on homology.

   737 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$

   738 and

   739 \eq{

   740     \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .

   741 }

   742 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.

   743 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.

   744 

   745 \medskip

   746 

   747 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob.

   748 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence.

   749 

   750 First, a lemma:  Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with

   751 $S^1$ replaced some (any) neighborhood of $* \in S^1$.

   752 Then $G''_*$ and $G'_*$ are both contractible

   753 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.

   754 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting

   755 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.

   756 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe

   757 in ``basic properties" section above} away from $*$.

   758 Thus any cycle lies in the image of the normal blob complex of a disjoint union

   759 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}).

   760 Actually, we need the further (easy) result that the inclusion

   761 $G''_* \to G'_*$ induces an isomorphism on $H_0$.

   762 

   763 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that

   764 for all $x \in F'_*$ we have

   765 \eq{

   766     x - \bd h(x) - h(\bd x) \in F''_* .

   767 }

   768 Since $F'_0 = F''_0$, we can take $h_0 = 0$.

   769 Let $x \in F'_1$, with single blob $B \sub S^1$.

   770 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$.

   771 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$).

   772 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$.

   773 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$.

   774 Define $h_1(x) = y$.

   775 The general case is similar, except that we have to take lower order homotopies into account.

   776 Let $x \in F'_k$.

   777 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$.

   778 Otherwise, let $B$ be the outermost blob of $x$ containing $*$.

   779 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$.

   780 So $x' \in G'_l$ for some $l \le k$.

   781 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$.

   782 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$.

   783 Define $h_k(x) = y \bullet p$.

   784 This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence.

   785 \nn{need to say above more clearly and settle on notation/terminology}

   786 

   787 Finally, we show that $F''_*$ is contractible.

   788 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}

   789 Let $x$ be a cycle in $F''_*$.

   790 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a

   791 ball $B \subset S^1$ containing the union of the supports and not containing $*$.

   792 Adding $B$ as a blob to $x$ gives a contraction.

   793 \nn{need to say something else in degree zero}

   794 

   795 This completes the proof that $F_*(C\otimes C)$ is

   796 homotopic to the 0-step complex $C$.

   797 

   798 \medskip

   799 

   800 Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$.

   801 $F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point *

   802 is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be.

   803 In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$.

   804 

   805 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows.

   806 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if

   807 * is a labeled point in $y$.

   808 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.

   809 Let $x \in \bc_*(S^1)$.

   810 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in

   811 $x$ with $y$.

   812 It is easy to check that $s$ is a chain map and $s \circ i = \id$.

   813 

   814 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points

   815 in a neighborhood $B_\ep$ of *, except perhaps *.

   816 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$.

   817 \nn{rest of argument goes similarly to above}

   818 

   819 \bigskip

   820 

   821 \nn{still need to prove exactness claim}

   822 

   823 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex?

   824 Do we need a map from hoch to blob?

   825 Does the above exactness and contractibility guarantee such a map without writing it

   826 down explicitly?

   827 Probably it's worth writing down an explicit map even if we don't need to.}

   828 

   829 

   830 We can also describe explicitly a map from the standard Hochschild

   831 complex to the blob complex on the circle. \nn{What properties does this

   832 map have?}

   833 

   834 \begin{figure}%

   835 $$\mathfig{0.6}{barycentric/barycentric}$$

   836 \caption{The Hochschild chain $a \tensor b \tensor c$ is sent to

   837 the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.}

   838 \label{fig:Hochschild-example}%

   839 \end{figure}

   840 

   841 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.

   842 The edges marked $x, y$ and $z$ carry the $1$-chains

   843 \begin{align*}

   844 x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\

   845 y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\

   846 z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab}

   847 \end{align*}

   848 and the $2$-chain labelled $A$ is

   849 \begin{equation*}

   850 A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}.

   851 \end{equation*}

   852 Note that we then have

   853 \begin{equation*}

   854 \bdy A = x+y+z.

   855 \end{equation*}

   856 

   857 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,

   858 $$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$

   859 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!)

   860