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## Frobenius algebras and 2D topological quantum field theories

by Joachim Kock

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

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).)

### Errata

• Page 75--77, 1.4.39: FLAW IN PROOF -- thanks to Chris Heunen and Jamie Vicary for pointing this out. There is a confusion between rectangular regions and connected components: the latter can be trapped inside each other without forming rectangular regions. A new proof strategy seems to be required. A correct proof (I believe) is given here.
• Page 33--34, Exercise 7-9: MATHEMATICAL ERROR -- thanks a lot to Jim Bryan for pointing it out. These three exercises are meaningless since the TQFT in question cannot exist. It corresponds to a non-commutative Frobenius algebra.
• Page 42, Proposition 1.3.23: MATHEMATICAL ERROR -- Thanks a lot to Chris Schommer-Pries and Gard Spreemann (independently) for pointing this out and correcting the error. The correct statement (Milnor, "Lectures on the h-Cobordism Theorem", Thm.1.9) is that two diffeomorphisms induce the same cobordism class iff they are pseudo-isotopic. (The notion of pseudo-isotopy is actually what was used in the proof of 1.3.23.)

Here is the replacement (for the proposition and its proof, as well as the paragraphs immediately before and after):  Two questions are natural at this point. Does every invertible cobordism arise from a diffeomorphism? When do two diffeomorphisms give the same cobordism class? The first question we will answer (affirmatively) only in the $2$-dimensional case, in the next section. The second question is settled by the next proposition. It relies on the notion of pseudo-isotopy: two diffeomorphisms $\psi_0 : \Sigma_0 \to \Sigma_1$ and $\psi_1 : \Sigma_0 \to \Sigma_1$ are said to be {\em pseudo-isotopic} if there is a diffeomorphism $\Psi: \Sigma_0 \times I \to \Sigma_1 \times I$ which agrees with $\psi_0$ in one end of the cylinder and with $\psi_1$ in the other: \begin{diagram}[w=6ex,h=3ex,tight] && \Sigma_1\times I && \\ & \ruTo^{(\psi_0,0)} && \luTo^{(\psi_1,1)} & \\ \Sigma_0 && \uTo>{\Psi} && \Sigma_0 \\ & \rdTo_{(\id,0)} && \ldTo_{(\id,1)} & \\ && \Sigma_0\times I &&&& \end{diagram} \begin{prop} Two diffeomorphisms $\Sigma_0 \topile \Sigma_1$ induce the same cobordism class $\Sigma_0 \cobto \Sigma_1$ if and only if they are pseudo-isotopic. \end{prop} \begin{dem} Suppose $\psi_0, \psi_1 : \Sigma_0 \topile \Sigma_1$ are pseudo-isotopic. Compose the above diagram with \begin{diagram}[w=6ex,h=4.5ex,tight] \Sigma_0 &\lTo^{\psi_1^{-1}} & \Sigma_1 \end{diagram} on the right, getting \begin{diagram}[w=6ex,h=3ex,tight] && \Sigma_1\times I && \\ & \ruTo^{(\psi_0,0)} && \luTo^{(\id,1)} & \\ \Sigma_0 && \uTo>{\Psi} && \Sigma_1 \\ & \rdTo_{(\id,0)} && \ldTo_{(\psi_1^{-1},1)} & \\ && \Sigma_0\times I &&&& \end{diagram} The upper part of this diagram is the cobordism class induced by $\psi_0$; the lower part is the cobordism induced by $\psi_1$ (in the backward' convention), and $\Psi$ expresses that they are equivalent. The converse implication amounts to reversing the argument. \end{dem} So in particular, a cobordism $\Sigma\cobto \Sigma$ induced by a diffeomorphism $\psi:\Sigma \isopil \Sigma$ is the identity if and only if $\psi$ is pseudo-isotopic to the identity. As an example of a diffeomorphism which is not pseudo-isotopic to the identity, take the twist diffeomorphism $\Sigma\disju \Sigma \to \Sigma\disju \Sigma$ which interchanges the two copies of $\Sigma$.
I should also take the opportunity to eliminate the abusive notation introduced in 1.3.22, and write $(\phi,0)$ instead of just $\phi$ in the diagram on page 46, etc, like in the above replacement.

• Page 51, line 26: 'interchanging label' should be 'interchanging labels'.
• Page 59, four lines from the bottom: there should be a blank line before the line that starts 'So instead...'.
• Page 68, line 4: the reference to [Hirsch] should be to 9.3.4, not to 4.4.2. Thanks to Marius Thaule for pointing it out.
• Page 70, last line:, the reference to 1.4.30 should rightly be to 1.4.32.
• Page 74, 1.4.37: The parenthesis "(Note that since the surface is connected, in fact every occurrence of [cap] must be to the left of a [mult]...)" should simply be deleted, as it is completely superfluous and actually wrong: it is possible to have a single [cap] and no [mult].
• Page 82, line -11: the second arrow should be \id_W \tensor \id_V \tensor \gamma_V instead of \id_W \tensor \id_V \tensor \gamma_W. (Gard Spreemann 2009-05-04 --- thanks!)
• Page 101: In the displayed formula in 2.2.19, there are two strange quote characters. They should not be there at all.
• Page 111: in the diamond shaped diagram, the two maps labelled 'mu' should rather be labelled 'beta'.
• Page 122, l.-7: The proof of Prop. 2.3.29 promises first to show that a certain map has 'epsilon' as counit, and satisfies the Frobenius equation. But the first promise is not fulfilled (thanks Carlos Moraga for pointing this out). There should be a sentence saying  That $\epsilon$ is a counit for $\delta \sigma$ follows by first using naturality of $\sigma$ with respect to $\epsilon$, and then the fact that $\epsilon$ is a counit for $\delta$.
• Page 130, line 1: it should be y^2+1 instead of y^2-1. (Thanks Carlos again.)
• Page 136, line 2: it should say k-algebra homomorphism instead of A-algebra homomorphism. (Thanks Carlos once again.)
• Page 136, line 8 (first diagram): The upper right-hand 'A' should be 'A \tensor A'. (This typo illustrates one advantage of graphical notation over symbolic: this sort of 'syntax error' could not possibly have occurred in graphical notation, like in the drawing just to the right of the diagram.)
• Page 153, last diagram in 3.2.10: in the right-hand side of the diagram, the vertical map should be from V'^n to V', and its name should be \mu'^(n).
• Page 155, line 7: It says: 'the result is not the same'. It should be 'the result is the same'.
• Page 163, line 13: 'that that' should be 'that'.
• Page 169, second line of Exercise 8: It says 'a family of maps'. It should be 'a family of invertible maps'.
• Page 174, second paragraph of 3.3.3 (thanks Travis Mandel 2018): it says that a cohmology ring is a graded-commutative Frobenius algebra. What this means is not clear: it is true that it is a graded-commutative algebra, meaning a commutative monoid in grVect. It is also true that it is a Frobenius algebra (cf. 2.2.23). But it is not a commutative Frobenius object in grVect, because the Frobenius form is not a morphism in grVect (as it is not of degree 0). Therefore, a cohomology ring does not define a 2D TQFT with values in grVect as claimed. The main message of 3.3.3 is still valid, though, namely that the symmetry condition is non-automatic. The cohomology example given to illustrate this can easily be fixed to illustrate the same point, by taking an even-dimensional manifold and Z/2Z-graded vector spaces. (Or more systematically, any n-dimensional manifold and Z/pZ-graded vector spaces for p|n.)
• Page 177, Exercise 7 and Exercise 8: MATHEMATICAL ERROR -- thanks a lot to Jim Bryan for pointing it out. The statement of the exercise 7 is false, and exercise 8 is therefore meaningless. I propose the following replacement exercises:  7. Consider now the $1$-dimensional Frobenius algebra $k$ over $k$, with Frobenius form $1 \mapsto u$. Show that the corresponding TQFT associates the invariant $u^{1-g}$ to a closed surface of genus $g$. Make sense of the formula also for nonconnected surfaces (hint: use 1.4.14). 8. Generalise the preceding exercise as follows: Let $(A,\epsilon)$ be a commutative Frobenius algebra, and let $a_g$ denote the invariant associated to a closed genus-$g$ surface by the corresponding TQFT. Show that if the Frobenius form $\epsilon$ is adjusted by a unit factor $u$ (cf.~Lemma 2.2.8), then the corresponding invariants $a_g$ will be adjusted by a factor $u^{1-g}$.
• Page 186, line 8: 'composition of face maps' should be replaced by 'composition of degeneracy maps'. (Thanks Carlos.)
• Page 235, in bib item [32], Lawvere's paper on ordinal sums, the publication year should be 1969, not 1967. (1967 is the year of the seminar from where the publication orginates.)

• I think it should be mentioned as Proposition 1.4.42 that assuming commutativity and cocommutativity as well as the naturality of the twist, then the two Frobenius equations imply each other. An extra exercise 6 (also in Section 1.4) should then be added asking to supply the proof, an instructive routine calculation.

### Remarks on the origin of the Frobenius equation

The 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 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 ---------------------------------------------------------------------- >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.) 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 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. `

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.