Free symmetric monoidal $(\infty,n)$-categories with duals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:14:29Z http://mathoverflow.net/feeds/question/49732 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49732/free-symmetric-monoidal-infty-n-categories-with-duals Free symmetric monoidal $(\infty,n)$-categories with duals domenico fiorenza 2010-12-17T14:31:08Z 2010-12-17T16:25:59Z <p>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. </p> <p>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).</p> <p>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?</p> http://mathoverflow.net/questions/49732/free-symmetric-monoidal-infty-n-categories-with-duals/49738#49738 Answer by Jacob Lurie for Free symmetric monoidal $(\infty,n)$-categories with duals Jacob Lurie 2010-12-17T16:25:59Z 2010-12-17T16:25:59Z <p>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>