| Summary: | <p>A fully extended framed topological field theory with target in a symmetric monoidal <em>n</em>-catgeory 𝐶 is a symmetric monoidal functor <em>Z</em> from Bord(n) to 𝐶, where Bord(n) is the symmetric monoidal <em>n</em>-category of <em>n</em>-framed bordisms. The cobordism hypothesis says that such field theories are classified by fully dualizable objects in 𝐶.</p> <p>Given a fully dualizable object <em>X</em> 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 <em>coherence data</em> for fully dualizable objects in a symmetric monoidal <em>n</em>-category. </p> <p>We prove a characterization of full dualizability of an object <em>X</em> in terms of existence of a dual of <em>X</em> 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 <em>n</em>=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.</p>
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