The reading of (Hopkins-)Lurie's On the Classification of Topological Field Theories (arXiv:0905.0465) suggests that a stronger version of the cobordism hypothesis should hold; namely, that (under eventually suitable technical assumptions), the inclusion of symmetric monoidal $(\infty,n)$-categories with duals into $(\infty,n)$-categories should have a left adjoint, the ``free symmetric monoidal $(\infty,n)$-category with duals on a given category $\mathcal{C}$ '', and that this free object should be given by a suitably $\mathcal{C}$-decorated $(\infty,n)$-cobordism $Bord_n(\mathcal{C})$. This would be an higher dimensional generalization of Joyal-Street-Reshetikhin-Turaev decorated tangles.

Such an adjunction would in particular give a canonical symmetric monoidal duality preserving functor $Z: Bord_n(\mathcal{C})\to \mathcal{C}$ which seems to appear underneath the constructions in Freed-Hopkins-Lurie-Teleman's Topological Quantum Field Theories from Compact Lie Groups (arXiv:0905.0731).

Yet, I've been unable to find an explicit statement of this conjectured adjointness in the above mentioned papers, and my google searches for "free symmetric monoidal n-category" only produce documents in which this continues with "generated by a single fully dualizable object", as in the original form of the cobordism hypothesis. Is anyone aware of a formal statement or treatment of the cobordism hypothesis from the left adjoint point of view hinted to above?

  • $\begingroup$ Could you give a reference for the 'Joyal-Street-Reshetikhin-Turaev decorated tangles' you're mentioning? I have a picture in mind of how to construct $Bord_1(\mathcal{C})$ explicitly and I'd like to compare it to the existing construction, but googling didn't give any results. (Sorry for 'excavating' this old question, but I think the idea is very interesting and it would be good to have a link or sth here for later reference. Thanks!) $\endgroup$ Mar 15, 2018 at 12:16
  • $\begingroup$ A nice description can be found in he second chapter of Bakalov-Kirillov's book "Lectures on Tensor Categories and Modular Functors". The name 'Joyal-Street-Reshetikhin-Turaev decorated tangles' is not explicitly used there: I used is as the formalism has been independently (as far as I know) introduced by Joyal-Street and by Reshetikhin-Turaev. $\endgroup$ Mar 15, 2018 at 15:04

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The existence of a left adjoint follows by formal nonsense. If you have a symmetric monoidal $(\infty,n)$-category which is can be built by first freely adjoining some objects, then some $1$-morphisms, then some $2$-morphisms, and so forth, up through $n$-morphisms and then stop, then there is an explicit geometric description of the $(\infty,n)$-category you get by "enforcing duality" in terms of manifolds with singularities. This is sketched in one of the sections of the paper you reference. I don't know of a geometric description for what you get if you start with an arbitrary symmetric monoidal $(\infty,n)$-category and then "enforce duality".

  • $\begingroup$ Yes, that's the section 4.3, Manifolds with singularities, from your paper on the classification of tqft, where singularity data are described. It is precisely that which suggested me the idea that those data were implicitly expressing the cobordism hypothesis as a left adjoint, but then I was unable to find this explicited in some form in the paper, so I became unsure about it. Thanks a lot. $\endgroup$ Dec 17, 2010 at 18:43
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    $\begingroup$ p.s. By the way, the existence of a natural morphisms $Bord^{SO}_n(\mathcal{C})\to \mathcal{C}$ for any symmetric monoidal $(\infty,n)$-category with duals seems to simplify the exposition of a few points in your paper with Freed-Hopkins-Teleman. Should you be interested in the details of this, there is an on-going forum discussion on the topic here: math.ntnu.no/~stacey/Mathforge/nForum/… $\endgroup$ Dec 17, 2010 at 18:48

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