On Nearly Frobenius Structures

Nearly Frobenius structures and 2-dimensional Almost TQFTs were introduced and shown to be in categorical equivalence in arXiv:1907.05470 in the attempt to extend the Atiyah-Segal's definition to the category of infinite dimensional vector spaces. In this paper, we investigate nearly Frobenius structures and we give a classification result for Almost TQFTs in dimension 2. In particular, the TQFTs functorial axioms become equivalent to the Edge Contraction/Construction axioms of colored ribbon graphs recently introduced by the authors.