497 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
497 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
498 \end{property} |
498 \end{property} |
499 |
499 |
500 \begin{property}[Contractibility] |
500 \begin{property}[Contractibility] |
501 \label{property:contractibility}% |
501 \label{property:contractibility}% |
502 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
502 The blob complex on an $n$-ball is contractible in the sense |
503 that it is homotopic to its $0$-th homology. |
503 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
504 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
504 \begin{equation*} |
505 associated by the system of fields $\cF$ to balls. |
505 \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
506 \begin{equation*} |
|
507 \xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
|
508 \end{equation*} |
506 \end{equation*} |
509 \end{property} |
507 \end{property} |
|
508 \nn{maybe should say something about the $A_\infty$ case} |
510 |
509 |
511 \begin{proof}(Sketch) |
510 \begin{proof}(Sketch) |
512 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
511 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
513 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
512 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
514 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
513 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
515 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
514 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
516 \end{proof} |
515 \end{proof} |
517 |
516 |
518 |
|
519 \subsection{Specializations} |
517 \subsection{Specializations} |
520 \label{sec:specializations} |
518 \label{sec:specializations} |
521 |
519 |
522 The blob complex has two important special cases. |
520 The blob complex has two important special cases. |
523 |
521 |
524 \begin{thm}[Skein modules] |
522 \begin{thm}[Skein modules] |
525 \label{thm:skein-modules} |
523 \label{thm:skein-modules} |
|
524 \nn{Plain n-categories only?} |
526 The $0$-th blob homology of $X$ is the usual |
525 The $0$-th blob homology of $X$ is the usual |
527 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
526 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
528 by $\cF$. |
527 by $\cC$. |
529 \begin{equation*} |
528 \begin{equation*} |
530 H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
529 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
531 \end{equation*} |
530 \end{equation*} |
532 \end{thm} |
531 \end{thm} |
|
532 This follows from the fact that the $0$-th homology of a homotopy colimit of \todo{something plain} is the usual colimit, or directly from the explicit description of the blob complex. |
533 |
533 |
534 \begin{thm}[Hochschild homology when $X=S^1$] |
534 \begin{thm}[Hochschild homology when $X=S^1$] |
535 \label{thm:hochschild} |
535 \label{thm:hochschild} |
536 The blob complex for a $1$-category $\cC$ on the circle is |
536 The blob complex for a $1$-category $\cC$ on the circle is |
537 quasi-isomorphic to the Hochschild complex. |
537 quasi-isomorphic to the Hochschild complex. |