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Contents

Idea

A cobordism category $Cob$ typically is canonically a dagger-category with dagger-involution given by reversal of the orientation of cobordisms.

Given a symmetric monoidal dagger category $\mathcal{C}$ as coefficients, a 1-functorial field theory

is called “unitary” if it is a dagger functor.

(This is the “hermitian axiom” of Atiyah 1989.)

References

The original notion:

• Michael Atiyah, p. 8 of: Topological quantum field theories, Publications Mathématiques de l’IHÉS 86 (1989) 175-186 [[numdam:PMIHES_1988__68__175_0](http://www.numdam.org/item?id=PMIHES_1988__68__175_0)]

• Vladimir Turaev, Hermitian and unitary TQFTs, §III.5 in: Quantum invariants of knots and 3-manifolds, Studies in Mathematics 18, de Gruyter (1994) [[doi:10.1515/9783110435221](https://doi.org/10.1515/9783110435221)]

Exposition and review:

• John Baez, p. 11 of Quantum Quandaries: a Category-Theoretic Perspective, in D. Rickles et al. (ed.) The structural foundations of quantum gravity, Clarendon Press (2006) 240-265 [[arXiv:quant-ph/0404040](https://arxiv.org/abs/quant-ph/0404040), ISBN:9780199269693]

Further discussion:

• Honglin Zhu, The Hermitian axiom on two-dimensional topological quantum field theories, J. Math. Phys. 64 (2023) 022301 [[arXiv:2206.07193](https://arxiv.org/abs/2206.07193), doi:10.1063/5.0121440]

On generalization to extended functorial field theory via higher dagger categories:

• Theo Johnson-Freyd, Higher Dagger Categories, talk at CQTS (08 Nov 2023) [slides:pdf, pdf]

See also:

• Lukas Müller, Luuk Stehouwer, Jan Steinebrunner, around Def 4.1 in: Reflection Structures and Spin Statistics in Low Dimensions [[arXiv:2301.06664](https://arxiv.org/abs/2301.06664)]