Coherence for 3-dualizable objects - Araújo, M (2017)
A fully extended framed topological field theory with target in a symmetric monoidal n-catgeory 𝐶 is a symmetric monoidal functor Z from Bord(n) to 𝐶, where Bord(n) is the symmetric monoidal n-category of n-framed bordisms. The cobordism hypothesis says that such field theories are classified by fully dualizable objects in 𝐶. Given a fully dualizable object X in 𝐶, we are interested in computing the values of the corresponding field theory on specific framed bordisms. This leads to the question of finding a presentation for Bord(n). In view of the cobordism hypothesis, this can be rephrased in terms of finding coherence data for fully dualizable objects in a symmetric monoidal n-category. We prove a characterization of full dualizability of an object X in terms of existence of a dual of X and existence of adjoints for a finite number of higher morphisms. This reduces the problem of finding coherence data for fully dualizable objects to that of finding coherence data for duals and adjoints. For n=3, and in the setting of strict symmetric monoidal 3-categories, we find this coherence data, and we prove the corresponding coherence theorems. The proofs rely on extensive use of a graphical calculus for strict monoidal 3-categories.