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". 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.)
At this Sussex meeting Ross Street reported on his discovery with Andre Joyal of braided monoidal categories (in the birth of which we also played a part - lecture by RFC Walters, Sydney Category Seminar, On a conversation with Aurelio Carboni and Bill Lawvere: the Eckmann-Hilton argument one-dimension up, 26th January 1983). This disovery was a major impulse towards the study of geometry and higher dimensional categories.
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. 136:497–529,1991
(submitted in 1988).
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 rather non-categorical assumption of an involution satisfying X^opp=X.
2004 Joachim Kock's book I jump to this because it enables me to discuss quickly later developments. The fronticepiece of the book
Joachim Kock, Frobenius algebras and 2D topological quantum field theories, Cambridge University Press 2004
is a picture of our equation S=X in the geometric form pointed out to us by Andre Joyal in 1987, the form which suggests the name S=X (maybe Z=X might be more suggestive but I prefer S=X). Kock discusses Lawvere's work on page 121, but, unaware of our introduction of the equation in 1985, he states that
"The first explicit mention of the Frobenius relation, and a proof of 2.3.4, were given in 1991 by Quinn" in
Frank Quinn, Lectures on axiomatic topoogical quantum field theory, in Geometry and quantum field theory, American Mathematical Society, 1995.
Kock's very pleasant book is a readable account of the relation between the algebra and the geometry of the Frobenius equation. It does not describe the physical intuition behind topological field theory, a concept introduced by Witten in 1988 in
Edward Witten, Topological quantum field theory, Communications in mathematical physics, 117, 353-38, 1988.

For another account of the history of category theory from an Australian point of view see the article
Ross Street, An Australian conspectus of higher categories, Institute for Mathematics and Applications Summer Program: n-categories: Foundations and Applications, June 2004.
Curiously, this article does not mention our discovery of the Frobenius equation.



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