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Book cover

Frobenius algebras and 2D topological quantum field theories

by Joachim Kock

xiv+240pp., No. 59 of LMSST, Cambridge University Press, 2003.

See also
This book, written for undergraduate math students, describes a striking connection between topology and algebra, expressed by the theorem that 2D topological quantum field theories are the same as commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The picture on the cover is the topological expression of the main axiom for a Frobenius algebra.
The book was reviewed in the Newsletter of the European Mathematical Society, March 2005. The reviewer (who signs himself mm) writes:
The book is very well written and organized. I warmly recommend it as an introduction to basic techniques of algebraic geometry.
This is a fantastic and highly surprising recommendation, since the book is not at all about algebraic geometry!

(A more serious review, by David Yetter, appeared in the Bulletin of the London Mathematical Society, 36 (2004).)



Remarks on the origin of the Frobenius equation

The Frobenius equation,

Frobenius equation

is the modern categorical characterisation of what it means to be a Frobenius algebra (Chapter 2), a characterisation that makes sense in any monoidal category, and hence more generally defines a notion of Frobenius object in any monoidal category. The category of 2-dimensional cobordisms (Chapter 1) is the free symmetric monoidal category on a commutative Frobenius object (Chapter 3). The classical characterisation of Frobenius algebra uses concepts like kernel and ideal that do not make sense outside a narrow abelian setting.

Bill Lawvere knew about the categorical characterisation of Frobenius algebras in 1967, but he did not explicitly write the Frobenius equation. In Chapter 2, I write that the first explicit appearance of the Frobenius equation is in the lecture notes of Quinn (published in 1995, lectures from 1991). This turns out to be wrong:

Aurelio Carboni and Bob Walters have pointed out to me (March 2006) that the first explicit appearance of the equation is in A. Carboni, R.F.C. Walters, Cartesian bicategories I, J. Pure Appl. Alg. 49 (1987), 11-32 (submitted February 1985). That paper studied the equation from another viewpoint (categories of relations), but without realising that it is also the equation characterising Frobenius algebras (in particular the authors were unaware of Lawvere's remark at the time). The pieces came together shortly after, according to the following historical account of the equation, which is very interesting and lively, and, it seems to me, very illustrative for the category theory community. I am grateful for their permission to reproduce it here. The text is also available from Bob Walter's Blog.

Date: Thu, 09 Mar 2006 14:01:25 +0100
From: RFC Walters
Subject: Some categorical history of the Frobenius equation
To: Joachim Kock
Cc: Aurelio Carboni

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
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
>From a posting in blog 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.)

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. 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
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.

The paragraph mentioning braided monoidal categories was not in the original email. It was added by Bob Walters a few days later. Bob also observes that Joyal's cobordism interpretation of Frobenius by 1987 must be seen in the context of the explosion of work on geometric interpretation of categorical equations (tangles, ribbons etc) that ensued from the advent of braided monoidal categories.

Last updated: 2014-11-09 by Joachim Kock.