A letter to Joachim Kock

Dear Joachim,
We have just been reading your very pleasant book about the relation between Frobenius algebras and cobordism. Perhaps you may be interested in some further history, from the categorical community, of the Frobenius equation, arising from a different line of research, and curiously not mentioned in the article by Ross Street, "An Australian conspectus of higher categories, Institute for Mathematics and Applications Summer Program, n-categories: Foundations and Applications, June, 2004".
One of us (Bob Walters) has written a blog entry (at
http://rfcwalters.blogspot.com)
recounting the story as we know it. We include that below.
As far as we know we were the first to explicitly publish the equation in 1987 (submitted February 1985), not Quinn as you report. But of course there may be even earlier occurrences, and there is the equivalent set of equations published by Lawvere in 1969. The other fact is that Joyal certainly knew the connection with cobordism when we talked with him in Louvain-la-Neuve in 1987.
best regards,
Aurelio Carboni and Robert FC Walters
Como, 9 March 2006
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From a posting in blog http://rfcwalters.blogspot.com Wednesday, February 15, 2006
History of an equation - (1 tensor delta)(nabla tensor 1)=(nabla)(delta)
This is a personal history of the equation
(1 tensor delta)(nabla tensor 1)=(nabla)(delta)
now called the Frobenius equation, or by computer scientists S=X.
1983 Milano:
Worked with Aurelio Carboni in Milano, and later in Sydney, on characterizing the category of relations.
1985 Sydney:
We submitted to JPAA on 12th February the paper eventually published as
A. Carboni, R.F.C. Walters, Cartesian bicategories I, Journal of Pure and Applied Algebra 49 (1987), pp. 11-32. The main equation was the Frobenius law, called by us discreteness or (D)(page 15).
1985 Isle of Thorns, Sussex:
Lectured on work with Carboni concentrating on importance of this new equation - replacing Freyd's "modular law" (see Freyd' book "Categories, Allegories"). Present in the audience were Joyal, Anders Kock, Lawvere, Mac Lane, Pitts, Scedrov, Street. I asked the audience to state the modular law, Joyal responded with the classical modular law, Pitts finally wrote the law on the board, but mistakenly. Scedrov said "So what?" to the new equation and "After all, the new law is equivalent to the modular law". Nobody ventured to have seen the equation before.
(I asked Freyd in Gummersbach in 1981 where he had found the modular law, and he replied that he found it by looking at all the small laws on relations involving intersection, composition and opposite, until he found the shortest one that generated the rest. We believe that this law actually occurs also in Tarski, A. Tarski, On the Calculus of Relations, J. of Symbolic Logic 6(3), pp. 73-89 (1941),
but certainly in the book "Set theory without variables" by Tarski and Givant, though not in the central role that Freyd emphasised.)
1987 Louvain-la-Neuve Conference:
I lectured on well-supported compact closed categories - every object has a structure satisfying the equation S=X, plus diamond=1. Aurelio spoke about his discovery that adding the axiom diamond=1 to the commutative and Frobenius equations characterizes commutative separable algebras, later reported in A. Carboni, Matrices, relations, and group representations, J. Alg. Vol 136, No 2,1991 (submitted in 1988)
(see in particular, the theorem and the remark in section 2). After Aurelio's lecture Andre Joyal stood up and declared that "These equations will never be forgotten!". At this, Sammy Eilenberg rather ostentatiously rose and left the lecture - perhaps the equation occurs already in Cartan-Eilenberg? Andre pointed out to us the geometry of the equation - drawing lots of 2-cobordisms. During the conference in a discussion in a bar with Joyal, Bill Lawvere and others, Bill recalled that he had written equations for Frobenius algebras in his work F.W. Lawvere, Ordinal Sums and Equational Doctrines, Springer Lecture
Notes in Mathematics No. 80, Springer-Verlag (1969), 141-155.
The equations did not incude S=X, diamond=1, or symmetry, but the equation S=X is easily deducible (see Carboni, "Matrices...", section 2). Bill's interest, as ours, was to discover a general notion of self-dual object. In Freyd's work there is instead the assumption of an involution satisfying X^opp=X, certainly a setting we dislike since it is categorically a nonsense, like saying that the dual of a vector space is equal to the space.