Frobenius algebras and 2D topological quantum field theories
by Joachim Kockxiv+240pp., No. 59 of LMSST, Cambridge University Press, 2003.
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Frobenius algebras and 2D topological quantum field theoriesby Joachim Kockxiv+240pp., No. 59 of LMSST, Cambridge University Press, 2003.
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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).)
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 $\Sigma$.
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| 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 $\delta$. |
| 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 $u^{1-g}$. |
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.