A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on Frobenius algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs in the unoriented case. This work provides a systematic study of extended Frobenius algebras in various settings: over a field, in a monoidal category, and in the framework of monoidal functors. Numerous examples, classification results, and general constructions of extended Frobenius algebras are established.
Keywords: extended Frobenius algebra, extended Frobenius monoidal functor, extended Hopf algebra
2020 MSC: 16K99, 18M15, 18M30, 57R56
Theory and Applications of Categories, Vol. 44, 2025, No. 36, pp 1218-1255.
Published 2025-11-07.
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