|
1 %!TEX root = ../blob1.tex |
|
2 |
1 In this section we analyze the blob complex in dimension $n=1$ |
3 In this section we analyze the blob complex in dimension $n=1$ |
2 and find that for $S^1$ the homology of the blob complex is the |
4 and find that for $S^1$ the blob complex is homotopy equivalent to the |
3 Hochschild homology of the category (algebroid) that we started with. |
5 Hochschild complex of the category (algebroid) that we started with. |
4 \nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
|
5 should perhaps put more emphasis on the complexes and less on the homology.} |
|
6 |
|
7 Notation: $HB_i(X) = H_i(\bc_*(X))$. |
|
8 |
|
9 Let us first note that there is no loss of generality in assuming that our system of |
|
10 fields comes from a category. |
|
11 (Or maybe (???) there {\it is} a loss of generality. |
|
12 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
|
13 thought of as the morphisms of a 1-category $C$. |
|
14 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
|
15 are $A(I; a, b)$, and composition is given by gluing. |
|
16 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
|
17 and neither does $A(I; a, b) = HB_0(I; a, b)$. |
|
18 But what about $HB_i(I; a, b)$ for $i > 0$? |
|
19 Might these higher blob homology groups be different? |
|
20 Seems unlikely, but I don't feel like trying to prove it at the moment. |
|
21 In any case, we'll concentrate on the case of fields based on 1-category |
|
22 pictures for the rest of this section.) |
|
23 |
|
24 (Another question: $\bc_*(I)$ is an $A_\infty$-category. |
|
25 How general of an $A_\infty$-category is it? |
|
26 Given an arbitrary $A_\infty$-category can one find fields and local relations so |
|
27 that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category? |
|
28 Probably not, unless we generalize to the case where $n$-morphisms are complexes.) |
|
29 |
|
30 Continuing... |
|
31 |
6 |
32 Let $C$ be a *-1-category. |
7 Let $C$ be a *-1-category. |
33 Then specializing the definitions from above to the case $n=1$ we have: |
8 Then specializing the definitions from above to the case $n=1$ we have: |
34 \begin{itemize} |
9 \begin{itemize} |
35 \item $\cC(pt) = \ob(C)$ . |
10 \item $\cC(pt) = \ob(C)$ . |
48 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
23 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
49 form $y - \chi(e(y))$. |
24 form $y - \chi(e(y))$. |
50 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
25 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
51 \end{itemize} |
26 \end{itemize} |
52 |
27 |
53 We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
28 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
54 Hochschild homology of $C$. |
29 Hochschild complex of $C$. |
55 \nn{Or better that the complexes are homotopic |
30 (Note that both complexes are free (and hence projective), so it suffices to show that they |
56 or quasi-isomorphic.} |
31 are quasi-isomorphic.) |
57 In order to prove this we will need to extend the blob complex to allow points to also |
32 In order to prove this we will need to extend the blob complex to allow points to also |
58 be labeled by elements of $C$-$C$-bimodules. |
33 be labeled by elements of $C$-$C$-bimodules. |
59 %Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product |
34 |
60 %(over $C$) of $C$-$C$-bimodules. |
|
61 %Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps. |
|
62 %Now we can define the blob complex for $S^1$. |
|
63 %This complex is the sum of complexes with a fixed cyclic tuple of bimodules present. |
|
64 %If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding |
|
65 %to the cyclic 1-tuple $(M)$. |
|
66 %In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled |
|
67 %by an element of $M$ and the remaining points are labeled by morphisms of $C$. |
|
68 %It's clear that $G_*(C)$ is isomorphic to the original bimodule-less |
|
69 %blob complex for $S^1$. |
|
70 %\nn{Is it really so clear? Should say more.} |
|
71 |
|
72 %\nn{alternative to the above paragraph:} |
|
73 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
35 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
74 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
36 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
75 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
37 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
76 other points. |
38 other points. |
77 The blob twig labels lie in kernels of evaluation maps. |
39 The blob twig labels lie in kernels of evaluation maps. |
78 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
40 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
79 Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
41 Let $K_*(M) = K_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
80 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
42 In other words, fields for $K_*(M)$ have an element of $M$ at the fixed point $*$ |
81 and elements of $C$ at variable other points. |
43 and elements of $C$ at variable other points. |
82 |
44 |
83 \todo{Some orphaned questions:} |
|
84 \nn{Or maybe we should claim that $M \to K_*(M)$ is the/a derived coend. |
|
85 Or maybe that $K_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild |
|
86 complex of $M$.} |
|
87 |
|
88 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
|
89 Do we need a map from hoch to blob? |
|
90 Does the above exactness and contractibility guarantee such a map without writing it |
|
91 down explicitly? |
|
92 Probably it's worth writing down an explicit map even if we don't need to.} |
|
93 |
|
94 |
45 |
95 We claim that |
46 We claim that |
96 \begin{thm} |
47 \begin{thm} |
97 The blob complex $\bc_*(S^1; C)$ on the circle is quasi-isomorphic to the |
48 The blob complex $\bc_*(S^1; C)$ on the circle is quasi-isomorphic to the |
98 usual Hochschild complex for $C$. |
49 usual Hochschild complex for $C$. |
99 \end{thm} |
50 \end{thm} |
100 |
|
101 \nn{Note that since both complexes are free (in particular, projective), |
|
102 quasi-isomorphic implies homotopy equivalent. |
|
103 This applies to the two claims below also. |
|
104 Thanks to Peter Teichner for pointing this out to me.} |
|
105 |
51 |
106 This follows from two results. First, we see that |
52 This follows from two results. First, we see that |
107 \begin{lem} |
53 \begin{lem} |
108 \label{lem:module-blob}% |
54 \label{lem:module-blob}% |
109 The complex $K_*(C)$ (here $C$ is being thought of as a |
55 The complex $K_*(C)$ (here $C$ is being thought of as a |
209 |
155 |
210 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
156 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
211 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
157 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
212 $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
158 $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
213 is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
159 is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
214 In other words, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
160 In particular, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
215 |
161 |
216 We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
162 We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
217 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
163 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
218 * is a labeled point in $y$. |
164 * is a labeled point in $y$. |
219 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
165 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
223 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
169 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
224 |
170 |
225 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
171 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
226 in a neighborhood $B_\ep$ of $*$, except perhaps $*$, and $B_\ep$ is either disjoint from or contained in every blob. |
172 in a neighborhood $B_\ep$ of $*$, except perhaps $*$, and $B_\ep$ is either disjoint from or contained in every blob. |
227 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
173 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
228 \nn{rest of argument goes similarly to above} |
|
229 |
174 |
230 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. |
175 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. |
231 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 |
176 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 |
232 of $x$ to $B_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
177 of $x$ to $B_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
233 write $y_i$ for the restriction of $z_i$ to $B_\ep$, and let |
178 write $y_i$ for the restriction of $z_i$ to $B_\ep$, and let |