506 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
506 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
507 \end{property} |
507 \end{property} |
508 |
508 |
509 \begin{property}[Contractibility] |
509 \begin{property}[Contractibility] |
510 \label{property:contractibility}% |
510 \label{property:contractibility}% |
511 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
511 The blob complex on an $n$-ball is contractible in the sense |
512 that it is homotopic to its $0$-th homology. |
512 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
513 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
513 \begin{equation*} |
514 associated by the system of fields $\cF$ to balls. |
514 \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)} |
515 \begin{equation*} |
|
516 \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)} |
|
517 \end{equation*} |
515 \end{equation*} |
518 \end{property} |
516 \end{property} |
|
517 \nn{maybe should say something about the $A_\infty$ case} |
519 |
518 |
520 \begin{proof}(Sketch) |
519 \begin{proof}(Sketch) |
521 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
520 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
522 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
521 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
523 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
522 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
524 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
523 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
525 \end{proof} |
524 \end{proof} |
526 |
525 |
527 |
|
528 \subsection{Specializations} |
526 \subsection{Specializations} |
529 \label{sec:specializations} |
527 \label{sec:specializations} |
530 |
528 |
531 The blob complex has two important special cases. |
529 The blob complex has two important special cases. |
532 |
530 |
533 \begin{thm}[Skein modules] |
531 \begin{thm}[Skein modules] |
534 \label{thm:skein-modules} |
532 \label{thm:skein-modules} |
|
533 \nn{Plain n-categories only?} |
535 The $0$-th blob homology of $X$ is the usual |
534 The $0$-th blob homology of $X$ is the usual |
536 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
535 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
537 by $\cF$. |
536 by $\cC$. |
538 \begin{equation*} |
537 \begin{equation*} |
539 H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
538 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
540 \end{equation*} |
539 \end{equation*} |
541 \end{thm} |
540 \end{thm} |
|
541 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. |
542 |
542 |
543 \begin{thm}[Hochschild homology when $X=S^1$] |
543 \begin{thm}[Hochschild homology when $X=S^1$] |
544 \label{thm:hochschild} |
544 \label{thm:hochschild} |
545 The blob complex for a $1$-category $\cC$ on the circle is |
545 The blob complex for a $1$-category $\cC$ on the circle is |
546 quasi-isomorphic to the Hochschild complex. |
546 quasi-isomorphic to the Hochschild complex. |