2 a few typos fixed

The reading of (Hopkins-)Lurie's On the Classification of Topological Field Theories (arXiv:0905.0465) suggests that a stronger version of the cobordism hypothesys 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 homotopy cobordism hypothesis from the left adjoint point of view hinted to above?

1

# 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 hypothesys 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 homotopy hypothesis from the left adjoint point of view hinted to above?