# Free symmetric monoidal $(\infty,n)$-categories with duals

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?

• 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!) – J. Steinebrunner Mar 15 '18 at 12:16
• 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. – domenico fiorenza Mar 15 '18 at 15:04

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".
• 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/… – domenico fiorenza Dec 17 '10 at 18:48