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What is the original reference where it was first proven that the generators and relations of the 2-dimensional cobordism category are those of commutative Frobenius algebras?

I've seen this article by Abrams being cited for it. But when I look into it I only find "Completeness of the relations follows easily by inspection" in proposition 12. I'm confused since I thought the completeness of relations was the only non-trivial part. The only place where I have seen something that looks like an actual proof is in this later article by Lauda and Pfeiffer in section 3.7, where they discuss 2-dimensional open-closed TQFT which obviously contains ordinary TQFT.

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3 Answers 3

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Have you looked at Joachim Kock's book "Frobenius algebras and 2D topological quantum field theories"?

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  • $\begingroup$ Yes, I found this one. I haven't really looked into it, but I'm sure it's done explicitly there. Is this the original reference though (it's rather new)? $\endgroup$
    – Andi Bauer
    Commented May 20, 2022 at 22:14
  • $\begingroup$ I meant that the original reference shall be in Kock's book. Though there might not be "the" orginial reference (see Kock's interesting "Remarks on the origin of the Frobenius equation" in the link I provided). $\endgroup$
    – DamienC
    Commented Jun 24, 2022 at 15:29
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Generators and relations for the nonextended 2-dimensional bordism category already appear in Robbert Dijkgraaf's 1989 PhD dissertation, see Section 3.2.

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Moore and Segal provide a list of references, but they also implicitly attribute it to folklore and/or Dan Friedan.

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