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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2011
    • (edited Feb 11th 2011)

    A while back I had a discussion here with Domenico on how the framed cobordim (,n)(\infty,n)-category Bord n fr(X)Bord^{fr}_n(X) of cobordisns in a topological space XX should be essentially the free symmetric monoidal (,n)(\infty,n)-category on the fundamental \infty-groupoid of XX.

    This can be read as saying

    Every flat \infty-parallel transport of fully dualizable objects has a unique \infty-holonomy.

    (!)

    Some helpful discussion with Chris Schommer-Pries tonight revealed that this is (unsurprisingly) already a special case of what Jacob Lurie proves. He proves it in more generality, which makes the statement easy to miss on casual reading. So I made it explicit now at cobordism hypothesis in the new section For cobordisms in a manifold.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2011

    I can flesh this out tomorrow. Am too tired now.

    • CommentRowNumber3.
    • CommentAuthordomenico_fiorenza
    • CommentTimeFeb 11th 2011
    • (edited Feb 11th 2011)

    Hi Urs,

    this is very good news. In view of this, sooner or later we should also look back at what we have been discussing on Topological Quantum Field Theories from Compact Lie Groups somewhere here on nForum.

  1. better sooner than later..

    Urs’ first post above can be rephrased as follows: TQFTs over a given oo-groupoid Π\Pi are the same thing as representations of Π\Pi on fully dualizable objects in an oo-symmetric category. in particular (fully extended) TQFTs over a manifold XX are the same thing of flat oo-bundles over XX. now, it would be interesting to have a push-forwarding of flat oo-bundles along morphisms XYX\to Y. by the above equivalence this would push TQFTs over XX forward to TQFTs over YY.

    Now, let us come to the case of a finite group GG. Here the relevant oo-groupoid Π\Pi is the delooping BG\mathbf{B}G, and a flat higher vector bundle over BG\mathbf{B}G is the same thing as an higher representation of GG. So we are in Dijkgraaf-Witten theory playground: a U(1)U(1)-valued 3-cocycle on GG, i.e. a morphism BGB 3U(1)\mathbf{B}G\to \mathbf{B}^3U(1) induces a “1-dimensional” 3-representation of GG, and so a TQFT over BG\mathbf{B}G. now we want to push this TQFT forward to a TQFT over the point. to do this we just have to push forward our 3-vector bundle from BG\mathbf{B}G to the point, i.e. to take its global sections.

    If Σ\Sigma is a manifold of dimension 3\leq 3 with a morphism to BG\mathbf{B}G, then we have an induced flat 3-bundle over Σ\Sigma and so a TQFT over Σ\Sigma. pushing this construction along H(Π(Σ),BG)*\mathbf{H}(\Pi(\Sigma),\mathbf{B}G)\to * should give the higher vector space associated to Σ\Sigma by push-forwarding the original TQFT on BG\mathbf{B}G to the point.

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