17 |
17 |
18 It is also worth noting that the original idea for the blob complex came from trying |
18 It is also worth noting that the original idea for the blob complex came from trying |
19 to find a more ``local" description of the Hochschild complex. |
19 to find a more ``local" description of the Hochschild complex. |
20 |
20 |
21 Let $C$ be a *-1-category. |
21 Let $C$ be a *-1-category. |
22 Then specializing the definitions from above to the case $n=1$ we have: |
22 Then specializing the definition of the associated system of fields from \S \ref{sec:example:traditional-n-categories(fields)} above to the case $n=1$ we have: |
23 \begin{itemize} |
23 \begin{itemize} |
24 \item $\cC(pt) = \ob(C)$ . |
24 \item $\cC(pt) = \ob(C)$ . |
25 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
25 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
26 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
26 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
27 points in the interior |
27 points in the interior |
42 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
42 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
43 Hochschild complex of $C$. |
43 Hochschild complex of $C$. |
44 In order to prove this we will need to extend the |
44 In order to prove this we will need to extend the |
45 definition of the blob complex to allow points to also |
45 definition of the blob complex to allow points to also |
46 be labeled by elements of $C$-$C$-bimodules. |
46 be labeled by elements of $C$-$C$-bimodules. |
47 (See Subsections \ref{moddecss} and \ref{ssec:spherecat} for a more general (i.e.\ $n>1$) |
47 (See Subsections \ref{moddecss} and \ref{ssec:spherecat} for a more general version of this construction that applies in all dimensions.) |
48 version of this construction.) |
|
49 |
48 |
50 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
49 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
51 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
50 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
52 The fields have elements of $M_i$ labeling |
51 The fields have elements of $M_i$ labeling |
53 the fixed points $p_i$ and elements of $C$ labeling other (variable) points. |
52 the fixed points $p_i$ and elements of $C$ labeling other (variable) points. |
77 \begin{lem} |
76 \begin{lem} |
78 \label{lem:module-blob}% |
77 \label{lem:module-blob}% |
79 The complex $K_*(C)$ (here $C$ is being thought of as a |
78 The complex $K_*(C)$ (here $C$ is being thought of as a |
80 $C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex |
79 $C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex |
81 $\bc_*(S^1; C)$. |
80 $\bc_*(S^1; C)$. |
82 (Proof later.) |
|
83 \end{lem} |
81 \end{lem} |
|
82 The proof appears below. |
84 |
83 |
85 Next, we show that for any $C$-$C$-bimodule $M$, |
84 Next, we show that for any $C$-$C$-bimodule $M$, |
86 \begin{prop} \label{prop:hoch} |
85 \begin{prop} \label{prop:hoch} |
87 The complex $K_*(M)$ is homotopy equivalent to $\HC_*(M)$, the usual |
86 The complex $K_*(M)$ is homotopy equivalent to $\HC_*(M)$, the usual |
88 Hochschild complex of $M$. |
87 Hochschild complex of $M$. |
286 for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$. |
285 for some $\widetilde{e_i} \in K$, and $\sum_i a_i \tensor e_i \tensor b_i = \hat{f}(\sum_i a_i \tensor \widetilde{e_i} \tensor b_i)$. |
287 Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ |
286 Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ |
288 such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$. |
287 such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$. |
289 For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$. |
288 For each $i$, we can find $\widetilde{q_i}$ so $g(\widetilde{q_i}) = q_i$. |
290 However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. |
289 However $\sum_i a_i \widetilde{q_i} b_i$ need not be zero. |
291 Consider then $$\widetilde{q} = \sum_i (a_i \tensor \widetilde{q_i} \tensor b_i) - 1 \tensor (\sum_i a_i \widetilde{q_i} b_i) \tensor 1.$$ Certainly |
290 Consider then $$\widetilde{q} = \sum_i \left(a_i \tensor \widetilde{q_i} \tensor b_i\right) - 1 \tensor \left(\sum_i a_i \widetilde{q_i} b_i\right) \tensor 1.$$ Certainly |
292 $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. |
291 $\widetilde{q} \in \ker(C \tensor E \tensor C \to E)$. |
293 Further, |
292 Further, |
294 \begin{align*} |
293 \begin{align*} |
295 \hat{g}(\widetilde{q}) & = \sum_i (a_i \tensor g(\widetilde{q_i}) \tensor b_i) - 1 \tensor (\sum_i a_i g(\widetilde{q_i}) b_i) \tensor 1 \\ |
294 \hat{g}(\widetilde{q}) & = \sum_i \left(a_i \tensor g(\widetilde{q_i}) \tensor b_i\right) - 1 \tensor \left(\sum_i a_i g(\widetilde{q_i}\right) b_i) \tensor 1 \\ |
296 & = q - 0 |
295 & = q - 0 |
297 \end{align*} |
296 \end{align*} |
298 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
297 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
299 |
298 |
300 Similar arguments show that the functors |
299 Similar arguments show that the functors |
418 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
417 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
419 Let $y_i$ be the restriction of $z_i$ to $N_\ep$. |
418 Let $y_i$ be the restriction of $z_i$ to $N_\ep$. |
420 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin N_\ep$, |
419 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin N_\ep$, |
421 and have an additional blob $N_\ep$ with label $y_i - s_\ep(y_i)$. |
420 and have an additional blob $N_\ep$ with label $y_i - s_\ep(y_i)$. |
422 Define $j_\ep(x) = \sum x_i$. |
421 Define $j_\ep(x) = \sum x_i$. |
423 \nn{need to check signs coming from blob complex differential} |
|
424 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
422 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
425 |
423 |
426 The key property of $j_\ep$ is |
424 The key property of $j_\ep$ is |
427 \eq{ |
425 \eq{ |
428 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep. |
426 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep. |
429 } |
427 } |
|
428 (Again, to get the correct signs, $N_\ep$ must be added as the first blob.) |
430 If $j_\ep$ were defined on all of $K_*(C\otimes C)$, this would show that $\sigma_\ep$ |
429 If $j_\ep$ were defined on all of $K_*(C\otimes C)$, this would show that $\sigma_\ep$ |
431 is a homotopy inverse to the inclusion $K'_* \to K_*(C\otimes C)$. |
430 is a homotopy inverse to the inclusion $K'_* \to K_*(C\otimes C)$. |
432 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
431 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
433 $\ep$ and show that $K'_*$ is homotopy equivalent to $K_*(C\otimes C)$. |
432 $\ep$ and show that $K'_*$ is homotopy equivalent to $K_*(C\otimes C)$. |
434 Instead, we'll be less ambitious and just show that |
433 Instead, we'll be less ambitious and just show that |
529 with |
528 with |
530 \eqar{ |
529 \eqar{ |
531 \bd(m\otimes a) & = & ma - am \\ |
530 \bd(m\otimes a) & = & ma - am \\ |
532 \bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a . |
531 \bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a . |
533 } |
532 } |
534 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.05}{hochschild/0-chains}$; the base point |
533 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.04}{hochschild/0-chains}$; the base point |
535 in $S^1$ is labeled by $m$ and there are no other labeled points. |
534 in $S^1$ is labeled by $m$ and there are no other labeled points. |
536 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
535 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
537 as shown in Figure \ref{fig:hochschild-1-chains}. |
536 as shown in Figure \ref{fig:hochschild-1-chains}. |
538 |
537 |
539 \begin{figure}[t] |
538 \begin{figure}[ht] |
540 \begin{equation*} |
539 \begin{equation*} |
541 \mathfig{0.4}{hochschild/1-chains} |
540 \mathfig{0.4}{hochschild/1-chains} |
542 \end{equation*} |
541 \end{equation*} |
543 \begin{align*} |
542 \begin{align*} |
544 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
543 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
545 \end{align*} |
544 \end{align*} |
546 \caption{The image of $m \tensor a$ in the blob complex.} |
545 \caption{The image of $m \tensor a$ in the blob complex.} |
547 \label{fig:hochschild-1-chains} |
546 \label{fig:hochschild-1-chains} |
548 \end{figure} |
547 \end{figure} |
549 |
548 |
550 \begin{figure}[t] |
549 \begin{figure}[ht] |
551 \begin{equation*} |
550 \begin{equation*} |
552 \mathfig{0.6}{hochschild/2-chains-0} |
551 \mathfig{0.6}{hochschild/2-chains-0} |
553 \end{equation*} |
552 \end{equation*} |
|
553 \caption{The 0-chains in the image of $m \tensor a \tensor b$.} |
|
554 \label{fig:hochschild-2-chains-0} |
|
555 \end{figure} |
|
556 \begin{figure}[ht] |
554 \begin{equation*} |
557 \begin{equation*} |
555 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
558 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
556 \end{equation*} |
559 \end{equation*} |
557 \caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$. |
560 \caption{The 1- and 2-chains in the image of $m \tensor a \tensor b$. |
558 Only the supports of the 1- and 2-blobs are shown.} |
561 Only the supports of the blobs are shown, but see Figure \ref{fig:hochschild-example-2-cell} for an example of a $2$-cell label.} |
559 \label{fig:hochschild-2-chains} |
562 \label{fig:hochschild-2-chains-12} |
560 \end{figure} |
563 \end{figure} |
561 |
564 |
562 \begin{figure}[t] |
565 \begin{figure}[ht] |
563 \begin{equation*} |
566 \begin{equation*} |
564 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
567 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
565 \end{equation*} |
568 \end{equation*} |
566 \begin{align*} |
569 \begin{align*} |
567 v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |
570 v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |
568 v_3 & = \mathfig{0.05}{hochschild/v_3-1} - \mathfig{0.05}{hochschild/v_3-2} & v_4 & = \mathfig{0.05}{hochschild/v_4-1} - \mathfig{0.05}{hochschild/v_4-2} |
571 v_3 & = \mathfig{0.05}{hochschild/v_3-1} - \mathfig{0.05}{hochschild/v_3-2} & v_4 & = \mathfig{0.05}{hochschild/v_4-1} - \mathfig{0.05}{hochschild/v_4-2} |
569 \end{align*} |
572 \end{align*} |
570 \caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains}.} |
573 \caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains-12}.} |
571 \label{fig:hochschild-example-2-cell} |
574 \label{fig:hochschild-example-2-cell} |
572 \end{figure} |
575 \end{figure} |
573 |
576 |
574 In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in |
577 In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in |
575 Figure \ref{fig:hochschild-2-chains}. |
578 Figures \ref{fig:hochschild-2-chains-0} and \ref{fig:hochschild-2-chains-12}. |
576 In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support. |
579 In Figure \ref{fig:hochschild-2-chains-12} the 1- and 2-blob diagrams are indicated only by their support. |
577 We leave it to the reader to determine the labels of the 1-blob diagrams. |
580 We leave it to the reader to determine the labels of the 1-blob diagrams. |
578 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
581 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
579 1-blob diagrams in its boundary. |
582 1-blob diagrams in its boundary. |
580 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
583 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
581 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
584 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
582 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell |
585 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell |
583 labeled $A$ in Figure \ref{fig:hochschild-2-chains}. |
586 labeled $A$ in Figure \ref{fig:hochschild-2-chains-12}. |
584 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
587 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
585 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
588 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
586 (Compare with the proof of \ref{bcontract}.) |
589 (Compare with the proof of \ref{bcontract}.) |