437 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
437 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
438 \end{property} |
438 \end{property} |
439 |
439 |
440 \begin{property}[Contractibility] |
440 \begin{property}[Contractibility] |
441 \label{property:contractibility}% |
441 \label{property:contractibility}% |
442 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
442 The blob complex on an $n$-ball is contractible in the sense |
443 that it is homotopic to its $0$-th homology. |
443 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
444 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
444 \begin{equation*} |
445 associated by the system of fields $\cF$ to balls. |
445 \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)} |
446 \begin{equation*} |
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447 \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)} |
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448 \end{equation*} |
446 \end{equation*} |
449 \end{property} |
447 \end{property} |
450 |
448 \nn{maybe should say something about the $A_\infty$ case} |
451 \nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
449 |
452 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
450 Properties \ref{property:functoriality}, \ref{property:disjoint-union} and \ref{property:gluing-map} are immediate from the definition. Property \ref{property:contractibility} \todo{} |
453 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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454 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
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455 |
451 |
456 \subsection{Specializations} |
452 \subsection{Specializations} |
457 \label{sec:specializations} |
453 \label{sec:specializations} |
458 |
454 |
459 The blob complex has two important special cases. |
455 The blob complex has two important special cases. |
460 |
456 |
461 \begin{thm}[Skein modules] |
457 \begin{thm}[Skein modules] |
462 \label{thm:skein-modules} |
458 \label{thm:skein-modules} |
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459 \nn{Plain n-categories only?} |
463 The $0$-th blob homology of $X$ is the usual |
460 The $0$-th blob homology of $X$ is the usual |
464 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
461 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
465 by $\cF$. |
462 by $\cC$. |
466 \begin{equation*} |
463 \begin{equation*} |
467 H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
464 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
468 \end{equation*} |
465 \end{equation*} |
469 \end{thm} |
466 \end{thm} |
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467 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. |
470 |
468 |
471 \begin{thm}[Hochschild homology when $X=S^1$] |
469 \begin{thm}[Hochschild homology when $X=S^1$] |
472 \label{thm:hochschild} |
470 \label{thm:hochschild} |
473 The blob complex for a $1$-category $\cC$ on the circle is |
471 The blob complex for a $1$-category $\cC$ on the circle is |
474 quasi-isomorphic to the Hochschild complex. |
472 quasi-isomorphic to the Hochschild complex. |