# August 6, 2008 {#week268}

This Week will be all about Frobenius algebras and modular tensor
categories. But first, here's a beautiful photo of Io, the volcanic
moon of Jupiter that I introduced back in ["Week 266"](#week266):

::: {align="center"}
[![](io_and_jupiter.jpg){width="600"}](io_and_jupiter.jpg)
:::

1) NASA Photojournal, A new year for Jupiter and Io,
`http://photojournal.jpl.nasa.gov/catalog/PIA02879`

Io looks awfully close to Jupiter here! It's actually 2.5 Jupiter
diameters away... but that's close enough to cause the intense tidal
heating that leads to sulfur volcanoes.

I told you about Frobenius algebras in ["Week 174"](#week174) and
["Week 224"](#week224), but I think it's time to talk about them
again! In the last few weeks, I've run into them - and their
generalizations - in a surprising variety of ways.

First of all, Jamie Vicary visited me here in Paris and explained how
certain Frobenius algebras can be viewed as classical objects living in
a quantum world - governed by quantum logic.

Mathematicians in particular are used to thinking of the quantum world
as a mathematical structure resting on foundations of classical logic:
first comes set theory, then Hilbert spaces on top of that. But what if
it's really the other way around? What if classical mathematics is
somehow sitting inside quantum theory? The world is quantum, after all.

There are a couple of papers so far that discuss this provocative idea:

2) Bob Coecke and Dusko Pavlovic, Quantum measurements without sums, in
The Mathematics of Quantum Computation and Technology, eds. Chen,
Kauffman and Lomonaco, Chapman and Hall/CRC, New York, pp. 559-596. Also
available as
[`arXiv:quant-ph/0608035`](http://arxiv.org/abs/quant-ph/0608035).

3) Jamie Vicary, Categorical formulation of quantum algebras, available
as [`arXiv:0805.0432`](http://arxiv.org/abs/0805.0432).

Second, Paul-André Melliès, the computer scientist and logician who's
my host here, has been telling me how logic can be nicely formulated in
certain categories - "*-autonomous categories" - which can be seen as
*categorified* Frobenius algebras. Here the idea goes back to Ross
Street:

4) Ross Street, Frobenius monads and pseudomonoids, J. Math. Physics 45
(2004) 3930-3948. Available as
`http://www.math.mq.edu.au/~street/Frob.pdf`

Paul-André is teaching a course on this and related topics; you can see
the slides for his course here:

5) Paul-André Melliès, Groupoides quantiques et logiques tensorielles:
une introduction, course notes at
`http://www.pps.jussieu.fr/~mellies/teaching.html`

See especially the fourth class.

But to get you ready for this material, I should give a quick
introduction to the basics!

If you're a normal mathematician, the easiest definition of "Frobenius
algebra" is something like this. For starters, it's an "algebra": a
vector space with an associative product that's linear in each
argument, and an identity element 1. But what makes it "Frobenius" is
that it's got a nondegenerate bilinear form g satisfying this axiom:

g(ab,c) = g(a,bc)

I'm calling it "g" to remind geometers of how nondegenerate bilinear
forms are used as "metrics", like the metric tensor at a point of a
Riemannian or Lorentzian manifold. But beware: we'll often work with
complex instead of real vector spaces. And, we won't demand that g(a,b)
= g(b,a), though this holds in many examples.

Let's see some examples! For starters, we could take the algebra of n \times
n matrices and define

g(a,b) = \operatorname{tr}(ab)

where "tr" is the usual trace. Or, we could perversely stick any
nonzero number in this formula, like

g(a,b) = -37 \operatorname{tr}(ab)

Or, we could take a bunch of examples like this and take their direct
sum. This gives us the most general "semisimple" Frobenius algebra.

So, semisimple Frobenius algebras are pathetically easy to classify.
There's also a vast wilderness of non-semisimple ones, which will never
be classified. But for a nice step in this direction, try Prop. 2 in
this paper:

6) Steve Sawin, Direct sum decompositions and indecomposable TQFTs, J.
Math. Phys. 36 (1995) 6673-6680. Also available as
[q-alg/9505026](http://arxiv.org/abs/q-alg/9505026).

This classifies all commutative Frobenius algebras that are
"indecomposable" - not a direct sum of others.

Note the mention of topological quantum field theories, or TQFTs.
Here's why. Suppose you have an $n$-dimensional TQFT. This gives vector
spaces for $(n-1)$-dimensional manifolds describing possible choices of
"space", and operators for $n$-dimensional manifolds going between
these, which describe possible choices of "spacetime".

So, it gives you some vector space for the $(n-1)$-sphere, say A. And,
this vector space is a commutative Frobenius algebra!

Let me sketch the proof. I'll use lots of hand-wavy reasoning, which is
easy to make rigorous using the precise definition of a TQFT.

For starters, there's the spacetime where two spherical universes
collide and fuse into one. Here's what it looks like for n = 2:

::: {align="center"}
![](multiplication.jpg)
:::

This gives the vector space A a multiplication:

    m\colon A \otimes A  \to A
       a \otimes b \mapsto ab

Next there's the spacetime where a spherical universe appears from
nothing - a "big bang":

::: {align="center"}
![](unit.jpg)
:::

This gives A an identity element, which we call 1:

    i\colon C  \to A
       1 \mapsto 1

Here C stands for the complex numbers, but mathematicians could use any
field.

Now we can use topology to show that A is an algebra - namely, that it
satisfies the associative law:

(ab)c = a(bc)

and the left and right unit laws:

1a = a = 1

::: {align="center"}
![](monoid_laws.jpg)
:::

But why is it a Frobenius algebra? To see this, let's switch the future
and past in our previous argument! The spacetime where a spherical
universe splits in two gives A a "comultiplication":

\Delta\colon A \to A \otimes A

::: {align="center"}
![](comultiplication.jpg)
:::

The spacetime where a spherical universe disappears into nothing - a
"big crunch" - gives A a trace, or more precisely a "counit":

e\colon A \to C

::: {align="center"}
![](counit.jpg)
:::

And, a wee bit of topology shows that these make A into a "coalgebra",
satisfying the "coassociative law" and the left and right "counit
laws":

::: {align="center"}
![](comonoid_laws.jpg)
:::

Everything has just been turned upside down!

It's easy to see that the multiplication on A is commutative, at least
for n > 1:

::: {align="center"}
![](commutative_law.jpg)
:::

Similarly, the comultiplication is "cocommutative" - just turn the
above proof upside down!

But why is A a Frobenius algebra? The point is that the algebra and
coalgebra structures interact in a nice way. We can use the product and
counit to define a bilinear form:

g(a,b) = e(ab)

This is just what we did in our matrix algebra example, where e was a
multiple of the trace.

We can also think of g as a linear operator

g\colon A \otimes A \to C

But now we see this operator comes from a spacetime where two universes
collide and then disappear into nothing:

::: {align="center"}
![](pairing.jpg)
:::

To check the Frobenius axiom, we just use associativity:

g(ab,c) = e((ab)c) = e(a(bc)) = g(a,bc)

::: {align="center"}
![](frobenius_pairing.jpg)
:::

But why is g nondegenerate? I'll just give you a hint. The bilinear
form g gives a map from A to the dual vector space A*:

a \mapsto g(a,-)

Physicists would call this map "lowering indices with the metric g".
To show that g is nondegenerate, it's enough to find an inverse for
this map, which physicists would call "raising indices". This should
be a map going back from A* to A. To build a map going back like this,
it's enough to get a map

h\colon C \to A \otimes A

and for this we use the linear operator coming from this spacetime:

::: {align="center"}
![](copairing.jpg)
:::

The fact that "raising indices" is the inverse of "lowering indices"
then follows from the fact that you can take a zig-zag in a piece of
pipe and straighten it out!

::: {align="center"}
![](zigzag.jpg)
:::

So, any $n$-dimensional TQFT gives a Frobenius algebra, and in fact a
commutative Frobenius algebra for n > 1.

In general there's more to the TQFT than this Frobenius algebra, since
there are spacetimes that aren't made of the building blocks I've
drawn. But in 2 dimensions, every spacetime can be built from these
building blocks: the multiplication and unit, comultiplication and
counit. So, with some work, one can show that

::: {align="center"}
A 2D TQFT IS THE SAME AS\
A COMMUTATIVE FROBENIUS ALGEBRA.
:::

This idea goes back to Dijkgraaf:

7) Robbert H. Dijkgraaf, A Geometric Approach To Two-Dimensional
Conformal Field Theory, PhD thesis, University of Utrecht, 1989.

and a formal proof was given by Abrams:

8) Lowell Abrams, Two-dimensional topological quantum field theories
and Frobenius algebra, Jour. Knot. Theory and its Ramifications 5
(1996), 569-587.

This book is probably the best place to learn the details:

9) Joachim Kock, Frobenius Algebras and 2d Topological Quantum Field
Theories, Cambridge U. Press, Cambridge, 2004.

but for a goofier explanation, try this:

10) John Baez, Winter 2001 Quantum Gravity Seminar, Track 1, weeks
11-17, `http://math.ucr.edu/home/baez/qg-winter2001/`

To prove the equivalence of 2d TQFTs and commutative Frobenius algebras,
it's handy to use a different definition of Frobenius algebra,
equivalent to the one I gave. I said a Frobenius algebra was an algebra
with a nondegenerate bilinear form satisfying

g(ab,c) = g(a,bc).

But this is equivalent to having an algebra that's also a coalgebra,
with multiplication and comultiplication linked by the "Frobenius
equations":

(\Delta \otimes 1~A~) (1~A~ \otimes m) = \Delta m = (m \otimes 1~A~) (1~A~ \otimes \Delta)

These equations are a lot more charismatic in pictures!

::: {align="center"}
![](frobenius_laws.jpg)
:::

We can also interpret them conceptually, as follows. If you have an
algebra A, it becomes an (A,A)-bimodule in an obvious way... well,
obvious if you know what this jargon means, at least. A \otimes A also becomes
an (A,A)-bimodule, like this:

a (b \otimes c) d = ab \otimes cd

Then, a Frobenius algebra is an algebra that's also a coalgebra, where
the comultiplication is an (A,A)-bimodule homomorphism! This scary
sentence has the Frobenius equations hidden inside it.

The Frobenius equations have a fascinating history, going back to
Lawvere, Carboni and Walters, Joyal, and others. Joachim Kock's website
includes some nice information about this. Read what Joyal said about
Frobenius algebras that made Eilenberg ostentatiously rise and leave the
room!

11) Joachim Kock, Remarks on the history of the Frobenius equation,
`http://mat.uab.es/~kock/TQFT.html#history`

The people I just mentioned are famous category theorists. They realized
that Frobenius algebra can be generalized from the category of vector
spaces to any "monoidal category" - that is, any category with tensor
products. And if this monoidal category is "symmetric", it has an
isomorphism between X \otimes Y and Y \otimes X for any objects X and Y, which lets
us generalize the notion of a *commutative* Frobenius object.

For a nice intro to these ideas, try the slides of this talk:

12) Ross Street, Frobenius algebras and monoidal category, talk at the
annual meeting of the Australian Mathematical Society, September 2004,
available at `http://www.maths.mq.edu.au/~street/FAMC.pdf`

These ideas allow for a very slick statement of the slogan I mentioned:

::: {align="center"}
A 2D TQFT IS THE SAME AS\
A COMMUTATIVE FROBENIUS ALGEBRA.
:::

For any n, there's a symmetric monoidal category nCob, with:

-   compact oriented $(n-1)$-manifolds as objects;
-   compact oriented $n$-dimensional cobordisms as morphisms.

The objects are choices of "space", and the morphisms are choices of
"spacetime".

The sphere is a very nice object in nCob; let's call it A. Then all the
pictures above show that A is a Frobenius algebra in nCob! It's
commutative when n > 1. And when n = 2, that's all there is to say!
More precisely:

::: {align="center"}
2Cob IS THE\
FREE SYMMETRIC MONOIDAL CATEGORY ON A COMMUTATIVE FROBENIUS ALGEBRA.
:::

So, to define a 2d TQFT, we just need to pick a commutative Frobenius
algebra in Vect (the category of vector spaces). By "freeness", this
determines a symmetric monoidal functor

Z\colon 2Cob \to Vect

and that's precisely what a 2d TQFT is!

If you don't know what a symmetric monoidal functor is, don't worry -
that's just what I'd secretly been using to translate from pictures of
spacetimes to linear operators in my story so far. You can get a precise
definition from those seminar notes of mine, or many other places.

Now let's talk about some variations on the slogan above.

We can think of the 2d spacetimes we've been drawing as the worldsheets
of "closed strings" - but ignoring the geometry these worldsheets
usually have, and keeping only the topology. So, some people call them
"topological closed strings".

We can also think about topological *open* strings, where we replace all
our circles by intervals. Just as the circle gave a commutative
Frobenius algebra, an interval gives a Frobenius algebra where the
multiplication comes from two open strings joining end-to-end to form a
single one:

::: {align="center"}
![](frobenius_algebra.jpg)
:::

This open string Frobenius algebra is typically noncommutative - draw
the picture and see! But, it's still "symmetric", meaning:

g(a,b) = g(b,a)

::: {align="center"}
![](symmetric_pairing_2.jpg)
:::

This is very nice. But physically, open strings like to join together
and form closed strings, so it's better to consider closed and open
strings together in one big happy family... or category.

The idea of doing this for topological strings was developed by Moore
and Segal:

13) Greg Moore, Lectures on branes, K-theory and RR charges, Clay Math
Institute Lecture Notes (2002), available at
`http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html`

Lauda and Pfeiffer developed this idea and proved that this category has
a nice description in terms of Frobenius algebras:

14) Aaron Lauda and Hendryk Pfeiffer, Open-closed strings:
two-dimensional extended TQFTs and Frobenius algebras, Topology Appl.
155 (2008) 623-666. Also available as
[arxiv:math.AT/0510664](http://arxiv.org/abs/math.AT/0510664).

Here's what they prove, encoded as a mysterious slogan:

::: {align="center"}
THE CATEGORY OF OPEN-CLOSED TOPOLOGICAL STRINGS IS THE\
FREE SYMMETRIC MONOIDAL CATEGORY\
ON A "KNOWLEDGEABLE" FROBENIUS ALGEBRA.\
:::

If you like the pictures I've been drawing so far, you'll *love* this
paper - since that's where I got most of these pictures! And, it's
just the beginning of a longer story where Lauda and Pfeiffer build 2d
TQFTs using state sum models:

15) Aaron Lauda and Hendryk Pfeiffer, State sum construction of
two-dimensional open-closed topological quantum field theories, J. Knot
Theory and its Ramifications 16 (2007), 1121-1163. Also available as
[`arXiv:math/0602047`](http://arXiv.org/abs/math/0602047).

This generalizes a construction due to Fukuma, Hosono and Kawai,
explained way back in ["Week 16"](#week16) and also in my seminar
notes mentioned above. Then Lauda and Pfeiffer use this machinery to
study knot theory!

16) Aaron Lauda and Hendryk Pfeiffer, Open-closed TQFTs extend Khovanov
homology from links to tangles, available as
[math/0606331](http://arXiv.org/abs/math/0606331).

Alas, explaining this would be a vast digression. I want to keep talking
about basic Frobenius stuff.

I guess I should say a bit more about semisimple versus non-semisimple
Frobenius algebras.

Way back at the beginning of this story, I said you can get a Frobenius
algebra by taking the algebra of n \times n matrices and defining

g(a,b) = k \operatorname{tr}(ab)

for any nonzero constant k. Direct sums of these give all the semisimple
Frobenius algebras.

But any algebra acts on itself by left multiplication:

L~a~: b \mapsto ab

so for any algebra we can try to define

g(a,b) = \operatorname{tr}(L~a~ L~b~)

This bilinear form is nondegenerate precisely when our algebra is
"strongly separable":

17) Marcelo Aguiar, A note on strongly separable algebras, available at
`http://www.math.tamu.edu/~maguiar/strongly.ps.gz`

Over the complex numbers, or any field of characteristic zero, an
algebra is strongly separable iff it's finite-dimensional and
semisimple. The story is trickier over other fields - see that last
paper of Lauda and Pfeiffer if you're interested.

Now, for n \times n matrices,

g(a,b) = \operatorname{tr}(L~a~ L~b~)

is n times the usual \operatorname{tr}(ab). But it's better, in a way. The reason is
that for any strongly separable algebra,

g(a,b) = \operatorname{tr}(L~a~ L~b~)

gives a Frobenius algebra with a cute extra property: if we comultiply
and then multiply, we get back where we started!

::: {align="center"}
![](separability.jpg)
:::

This is easy to see if you write the above formula for g using diagrams.
Frobenius algebras with this cute extra property are sometimes called
"special".

If we use a commutative special Frobenius algebra to get a 2d TQFT, it
fails to detect handles! That seems sad. But these papers:

18) Stephen Lack, Composing PROPs, Theory and Applications of
Categories 13(2004), 147-163. Available at
`http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html`

19) R. Rosebrugh, N. Sabadini and R.F.C. Walters, Generic commutative
separable algebras and cospans of graphs, Theory and Applications of
Categories 15 (Proceedings of CT2004), 164-177. Available at
`http://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html`

makes that sad fact seem good! Namely:

::: {align="center"}
Cospan(\mathsf{FinSet}) IS THE\
FREE SYMMETRIC MONOIDAL CATEGORY\
ON A COMMUTATIVE SPECIAL FROBENIUS ALGEBRA.
:::

Here Cospan(\mathsf{FinSet}) is the category of "cospans" of finite sets. The
objects are finite sets, and a morphism from X to Y looks like this:

                      X         Y
                       \       /
                       F\     /G
                         \   /
                          v v
                           S

If you remember the "Tale of Groupoidication" starting in
["Week 247"](#week247), you'll know about spans and how to compose
spans using pullback. This is just the same only backwards: we compose
cospans using pushout.

But here's the point. A 2d cobordism is itself a kind of cospan:

                      X         Y
                       \       /
                       F\     /G
                         \   /
                          v v
                           S

with two collections of circles included in the 2d manifold S. If we
take connected components, we get a cospan of finite sets. Now we've
lost all information about handles! And the circle - which was a
commutative Frobenius algebra - becomes a mere one-point set - which is
a *special* commutative Frobenius algebra.

Now for a few examples of *non*-semisimple Frobenius algebras.

First, take the exterior algebra ΛV over an $n$-dimensional vector space
V, and pick any nonzero element of degree n - what geometers would call
a "volume form". There's a unique linear map

e\colon ΛV \to C

which sends the volume form to 1 and kills all elements of degree < n.
This is a lot like "integration" - and so is taking a trace. So, you
should want to make ΛV into a Frobenius algebra using this formula:

g(a,b) = e(a ^ b)

where ^ is the product in the exterior algebra. It's easy to see this
is nondegenerate and satisfies the Frobenius axiom:

g(ab,c) = e(a ^ b ^ c) = g(a,bc)

So, it works! But, this algebra is far from semisimple.

If you know about cohomology, you should want to copy this trick
replacing the exterior algebra by the deRham cohomology of a compact
oriented manifold, and replacing e by "integration". It still works.
So, every compact manifold gives us a Frobenius algebra!

If you know about algebraic varieties, you might want to copy *this*
trick replacing the compact manifold by a complex projective variety.
I'm no expert on this, but people seem to say that it only works for
Calabi-Yau varieties. Then you can do lots of cool stuff:

20) Kevin Costello, Topological conformal field theories and Calabi-Yau
categories, available as
[arxiv:math/0412149](http://arxiv.org/abs/math/0412149).

Here a "Calabi-Yau category" is just the "many-object" version of a
Frobenius algebra - a Calabi-Yau category with one object is a Frobenius
algebra. There's much more to say about this wonderful paper, but I'm
afraid for now you'll have to read it... I'm getting worn out, and I
want to get to the new stuff I just learned!

But before I do, I can't resist rounding off one corner I cut. I said
that Frobenius algebras show up naturally by taking string theory and
watering it down: ignoring the geometrical structure on our string
worldsheets and remembering only their topology. A bit more precisely,
2d TQFTs assign linear operators to 2d cobordisms, but *conformal* field
theories assign operators to 2d cobordisms *equipped with conformal
structures*. Can we describe conformal field theories using Frobenius
algebras?

Yes!

21) Ingo Runkel, Jens Fjelstad, Jurgen Fuchs, Christoph Schweigert,
Topological and conformal field theory as Frobenius algebras, available
as [`arXiv:math/0512076`](http://arXiv.org/abs/math/0512076).

But, you need to use Frobenius algebras inside a modular tensor
category!

I wish I had more time to study modular tensor categories, and tell you
all about them. They are very nice braided monoidal categories that are
*not* symmetric. You can use them to build 3d topological quantum field
theories, and they're also connected to other branches of math.

For example, you can modular tensor categories consisting of nice
representations of quantum groups. You can also can get them from
rational conformal field theories - which is what the above paper by
Runkel, Fjelstad, Fuchs and Schweigert is cleverly turning around. You
can also get them from von Neumann algebras!

If you want to learn the basics, this book is great - there's a
slightly unpolished version free online:

22) B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and
Modular Functors, American Mathematical Society, Providence, Rhode
Island, 2001. Preliminary version available at
`http://www.math.sunysb.edu/~kirillov/tensor/tensor.html`

But if a book is too much for you, here's a nice quick intro. It
doesn't say much about topological or conformal field theory, but it
gives a great overview of recent work on the algebraic aspects of tensor
categories:

23) Michael Müger, Tensor categories: a selective guided tour,
available as [`arXiv:0804.3587`](http://arxiv.org/abs/0804.3587).

Here's a quite different introduction to recent developments, at least
up to 2004:

24) Damien Calaque and Pavel Etingof, Lectures on tensor categories,
available as
[`arXiv:math/0401246`](http://arXiv.org/abs/arXiv:math/0401246).

Still more recently, Hendryk Pfeiffer has written what promises to be a
fundamental paper describing how to think of any modular tensor category
as the category of representations of an algebraic gadget - a "weak
Hopf algebra":

25) Hendryk Pfeiffer, Tannaka-Krein reconstruction and a
characterization of modular tensor categories, available as
[`arXiv:0711.1402`](http://arxiv.org/abs/0711.1402).

And here's a paper that illustrates the wealth of examples:

26) Seung-moon Hong, Eric Rowell, Zhenghan Wang, On exotic modular
tensor categories, available as
[`arXiv:07108.5761`](http://arxiv.org/abs/0710.5761).

The abstract of this makes me realize that people have bigger hopes of
understanding all modular tensor categories than I'd imagined:

> It has been conjectured that every (2+1)-dimensional TQFT is a
> Chern-Simons-Witten (CSW) theory labelled by a pair (G,k), where G is
> a compact Lie group, and k in H^4(BG,Z) is a cohomology class. We
> study two TQFTs constructed from Jones' subfactor theory which are
> believed to be counterexamples to this conjecture: one is the quantum
> double of the even sectors of the \mathrm{E}_6 subfactor, and the other is the
> quantum double of the even sectors of the Haagerup subfactor. We
> cannot prove mathematically that the two TQFTs are indeed
> counterexamples because CSW TQFTs, while physically defined, are not
> yet mathematically constructed for every pair (G,k). The cases that
> are constructed mathematically include:
>
> -   G is a finite group - the Dijkgraaf-Witten TQFTs;
> -   G is a torus T^n;
> -   G is a connected semisimple Lie group - the Reshetikhin-Turaev
>     TQFTs.
>
> We prove that the two TQFTs are not among those mathematically
> constructed TQFTs or their direct products. Both TQFTs are of the
> Turaev-Viro type: quantum doubles of spherical tensor categories. We
> further prove that neither TQFT is a quantum double of a braided
> fusion category, and give evidence that neither is an orbifold or
> coset of TQFTs above. Moreover, the representation of the braid groups
> from the half \mathrm{E}_6 TQFT can be used to build universal topological
> quantum computers, and the same is expected for the Haagerup case.

Anyway, now let me say what Vicary and Melliès have been explaining to
me. I'll give it in a highly simplified form... and all mistakes are
my own.

First, from what I've said already, every commutative special Frobenius
algebra over the complex numbers looks like

C \oplus  C \oplus  ... C \oplus  C

It's a direct sum of finitely many copies of C, equipped with its
god-given bilinear form

g(a,b) = \operatorname{tr}(L~a~ L~b~)

So, this sort of Frobenius algebra is just an algebra of complex
functions on a *finite set*. A map between finite sets gives an algebra
homomorphism going back the other way. And the algebra homomorphisms
between two Frobenius algebras of this sort *all* come from maps between
finite sets.

So, the category with:

-   commutative special complex Frobenius algebras as objects;
-   algebra homomorphisms as morphisms

is equivalent to \mathsf{FinSet}^op^. This means we can find the category of
finite sets - or at least its opposite, which is just as good - lurking
inside the world of Frobenius algebras!

Coecke, Pavlovic and Vicary explore the ramifications of this result for
quantum mechanics, using Frobenius algebras that are Hilbert spaces
instead of mere vector spaces. This lets them define a "†-Frobenius
algebra" to be one where the comultiplication and counit are adjoint to
the the multiplication and unit. They show that making a
finite-dimensional Hilbert space into a commutative special †-Frobenius
algebra is the same as *equipping it with an orthonormal basis*.

There's no general way to duplicate quantum states - "you can't clone
a quantum" - but if you only want to duplicate states lying in a chosen
orthonormal basis you can do it. So, you can think of commutative
special †-Frobenius algebras as "classical data types", which let you
duplicate information. That's what the comultiplication does:
duplicate!

::: {align="center"}
![](comultiplication.jpg)
:::

Any commutative special †-Frobenius algebra has a finite set attached to
it: namely, the set of basis elements. So, we now see how to describe
finite sets starting from Hilbert spaces and introducing a notion of
"classical data type" formulated purely in terms of quantum concepts.

The papers by Coecke, Pavlovic and Vicary go a lot further than my
summary here. Jamie Vicary even studies how to *categorify* everything
I've just mentioned!

A subtlety: it's a fun puzzle to show that in any monoidal category,
morphisms between Frobenius algebras that preserve *all* the Frobenius
structure are automatically *isomorphisms*. See the slides of Street's
talk if you get stuck: he shows how to construct the inverse, but you
still get the fun of proving it works.

So, the category with:

-   commutative special complex Frobenius algebras as objects;
-   Frobenius homomorphisms as morphisms

is equivalent to the *groupoid* of finite sets. We get \mathsf{FinSet}^op^ if we
take algebra homomorphisms, and I guess we get \mathsf{FinSet} if we take
coalgebra homomorphisms.

Finally, a bit about categorified Frobenius algebras and logic!

I'm getting a bit tired, so I hope you believe that the concept of
Frobenius algebra can be categorified. As I already mentioned, Frobenius
algebras make sense in any monoidal category - and then they're
sometimes called "Frobenius monoids". Similarly, categorified
Frobenius algebras make sense in any monoidal bicategory, and then
they're sometimes called "Frobenius pseudomonoids". These were
introduced in Street's paper "Frobenius monads and pseudomonoids",
cited above - but if you like pictures, you may also enjoy learning
about them here:

27) Aaron Lauda, Frobenius algebras and ambidextrous adjunctions,
Theory and Applications of Categories 16 (2006), 84-122, available at
`http://tac.mta.ca/tac/volumes/16/4/16-04abs.html`\
Also available as
[`arXiv:math/0502550`](http://arXiv.org/abs/math/0502550).

I explained some of the basics behind this paper in
["Week 174"](#week174).

But now, I want to give a definition of *-autonomous categories, which
simultaneously makes it clear that they're natural structures in logic,
and that they're categorified Frobenius algebras!

Suppose A is any category. We'll call its objects "propositions" and
its morphisms "proofs". So, a morphism

f\colon a \to b

is a proof that a implies b.

Next, suppose A is a symmetric monoidal category and call the tensor
product "or". So, for example, given proofs

f: a \to b, f'\colon a' \to b'

we get a proof

f or f'\colon a or a' \to b or b'

Next, suppose we make the opposite category A^op^ into a symmetric
monoidal category, but with a completely different tensor product, that
we'll call "and". And suppose we have a monoidal functor:

not\colon A \to A^op^

So, for example, we have

not(a or b) = not(a) and not(b)

or at least they're isomorphic, so there are proofs going both ways.

Now we can apply "op" and get another functor I'll also call "not":

not\colon A^op^ \to A

Using the same name for this new functor could be confusing, but it
shouldn't be. It does the same thing to objects and morphisms; we're
just thinking about the morphisms as going backwards.

Next, let's demand that this new functor be monoidal! This too is quite
reasonable; for example it implies that

not(a and b) = not(a) or not(b)

or at least they're isomorphic.

Next, let's demand that this pair of functors:

          not
       --------->
    A             Aop
      <----------
          not

be a monoidal adjoint equivalence. So, for example, there's a
one-to-one correspondence between proofs

not(a) \to b

and proofs

not(b) \to a

Now for the really fun part. Let's define a kind of "bilinear form":

g\colon A \times A \to \mathsf{Set}

where g(a,b) is the set of proofs

not(a) \to b

And let's demand that g satisfy the Frobenius axiom! In other words,
let's suppose there's a natural isomorphism:

g(a or b, c) \cong g(a, b or c)

Then A is a "*-autonomous category"! And this is a sensible notion,
since it amounts to requiring a natural one-to-one correspondence
between proofs

not(a or b) \to c

and proofs

not(a) \to b or c

So, categorified Frobenius algebras are a nice framework for
propositional logic!

In case it slipped by too fast, let me repeat the definition of
*-autonomous category I just gave. It's a symmetric monoidal category
A with a monoidal adjoint equivalence called "not" from A (with one
tensor product, called "or") to A^op^ (with another, called "and"),
such that the functor

    g\colon A \times A  \to \mathsf{Set}
       (a,b) \mapsto \operatorname{Hom}(not(a),b)

is equipped with a natural isomorphism

g(a or b, c) \cong g(a, b or c)

I hope I didn't screw up. I want this definition to be equivalent to
the usual one, which was invented by Michael Barr quite a while ago:

28) Michael Barr, *-Autonomous Categories, Lecture Notes in
Mathematics 752, Springer, Berlin, 1979.

By now *-autonomous categories become quite popular among those working
at the interface of category theory and logic. And, there are many ways
to define them. Brady and Trimble found a nice one:

29) Gerry Brady and Todd Trimble, A categorical interpretation of C. S.
Peirce's System Alpha, Jour. Pure Appl. Alg. 149 (2000), 213-239.

Namely, they show a *-autonomous category is the same as a symmetric
monoidal category A equipped with a *contravariant* adjoint equivalence

not\colon A \to A

which is equipped with a "strength", and where the unit and counit of
the adjunction respect this strength.

Later, in his paper "Frobenius monads and pseudomonoids", Street
showed that *-autonomous categories really do give Frobenius
pseudomonoids in a certain monoidal bicategory with:

-   categories as objects;
-   profunctors (also known as distributors) as morphisms;
-   natural transformations as $2$-morphisms.

Alas, I'm too tired to explain this now! It's a slicker way of saying
what I already said. But the cool part is that this bicategory is like a
categorified version of Vect, with the category of finite sets replacing
the complex numbers. That's why in logic, the "nondegenerate bilinear
form" looks like

g\colon A \times A \to \mathsf{Set}

So: Frobenius algebras are lurking all over in physics, logic and
quantum logic, in many tightly interconnected ways. There should be some
unified explanation of what's going on! Do you have any ideas?

Finally, here are two books on math and music that I should read
someday. The first seems more elementary, the second more advanced:

30) Trudi Hammel Garland and Charity Vaughan Kahn, Math and Music -
Harmonious Connections, Dale Seymour Publications, 1995. Review by
Elodie Lauten on her blog Music Underground,
`http://www.sequenza21.com/2007/04/microtonal-math-heads.html`

31) Serge Donval, Histoire de l'Acoustique Musicale (History of
Musical Acoustics), Editions Fuzaeau, Bressuire, France, 2006. Review at
Music Theory Online,
`http://mto.societymusictheory.org/mto-books.html?id=11`

------------------------------------------------------------------------

**Addenda**: I thank Bob Coecke, Robin Houston, Steve Lack, Paul-André
Melliès, Todd Trimble, Jamie Vicary, and a mysterious fellow named
Stuart for some very helpful corrections.

You can't really appreciate the pictorial approach to Frobenius
algebras until you use it to prove some things. Try proving that every
homomorphism of Frobenius algebras is an isomorphism! Or for something
easier, but still fun, start by assuming that a Frobenius algebra is an
algebra and coalgebra satisfying the Frobenius equations

::: {align="center"}
![](frobenius_laws.jpg)
:::

and use this to prove the following facts:

::: {align="center"}
![](comultiplication_facts.jpg)
:::

For more discussion, visit the [$n$-Category Café](http://golem.ph.utexas.edu/category/2008/08/this_weeks_finds_in_mathematic_29.html).
In particular, you'll see there's a real morass of conflicting
terminology concerning what I'm calling "special" Frobenius algebras
and "strongly separable" algebras. But if we define them as I do
above, they're very nicely related.

More precisely: an algebra is strongly separable iff it can be given a
comultiplication and counit making it into a special Frobenius algebra.
If we can do this, we can do it in a unique way. Conversely, the
underlying algebra of a special Frobenius algebra is strongly separable.

For more details, see:

32) nLab, Frobenius algebra,
`http://ncatlab.org/nlab/show/Frobenius+algebra`

and:

33) nLab, Separable algebra,
`http://ncatlab.org/nlab/show/separable+algebra`

------------------------------------------------------------------------

*'Interesting Truths' referred to a kind of theorem which captured
subtle unifying insights between broad classes of mathematical
structures. In between strict isomorphism - where the same structure
recurred exactly in different guises - and the loosest of poetic
analogies, Interesting Truths gathered together a panoply of apparently
disparate systems by showing them all to be reflections of each other,
albeit in a suitably warped mirror.* - Greg Egan, *Incandescence*