Baez–Dolan cobordism hypothesis for framed extended topological field theories

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Let C\mathcal{C} be a symmetric monoidal (∞,n)(\infty,n)-category with duals. A framed extended nn-dimensional TFT valued in C\mathcal{C} is a symmetric monoidal functor from the framed bordism (∞,n)(\infty,n)-category to C\mathcal{C}, and C∼\mathcal{C}^{\sim} denotes its fully dualizable subcategory. Baez–Dolan's framed cobordism hypothesis. The evaluation functor Z↦Z(∗)Z\mapsto Z(*) induces an equivalence

Fun⁡⊗(Bordnfr,C)≃C∼\operatorname{Fun}^{\otimes}(\mathbf{Bord}_n^{\mathrm{fr}},\mathcal{C})\simeq\mathcal{C}^{\sim}

between framed extended nn-dimensional TFTs valued in C\mathcal{C} and the fully dualizable subcategory of C\mathcal{C}. This conjecture asserts that a framed extended field theory is completely determined by its value on a point; the supplied text gives no evidence of resolution, so its status remains open.

References

Primary source

Araminta Amabel, “Notes on Factorization Algebras and TQFTs”, arXiv:2307.01306 (2023).

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