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August 6, 2008

This Week's Finds in Mathematical Physics (Week 268)

John Baez

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 "week266":

1) JPL 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 "week174" and "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, to appear in The Mathematics of Quantum Computation and

Technology, eds. Chen, Kauffman and Lomonaco. Also available as

arXiv:quant-ph/0608035.

3) Jame Vicary, Categorical formulation of quantum algebras,

available as arXiv:0805.0432.

Second, Paul-Andre Mellies, 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 pseudomonads, J. Math. Physics

45 (2004) 3930-3948. Available as

http://www.math.mq.edu.au/~street/Frob.pdf

Paul-Andre is teaching a course on this and related topics; you

can see the slides for his course here:

5) Paul-Andre Mellies, 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 x n matrices and define

g(a,b) = tr(ab)

where "tr" is the usual trace. Or, we could perversely stick

any nonzero number in this formula, like

g(a,b) = -37 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.

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:

______ ______

/ \ / \

| | | |

| | | |

| | | |

|\______/| |\______/|

| | | |

| | | |

\ \ / /

\ \ / /

\ \_/ /

\ /

\ /

\ /

\ /

| |

| |

| ....... |

|' `|

| |

| |

\ /

\_______/

This gives the vector space A a multiplication:

m: A tensor A -> A

a tensor b |-> ab

Next there's the spacetime where a spherical universe appears

from nothing - a "big bang":

_____

/ \

/ \

| |

| |

| |

| |

| ....... |

|' `|

| |

| |

\ /

\_______/

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

i: C -> A

1 |-> 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

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: A -> A tensor A

_______

/ \

| |

| |

| |

|\_______/|

| |

| |

| |

| |

/ \

/ \

/ \

/ \

/ _ \

/ / \ \

/..... / \ .....\

/ ` / \ ' \

/ `/ \' \

| | | |

| | | |

\ / \ /

\______/ \______/

The spacetime where a spherical universe disappears into nothing -

a "big crunch" - gives A a trace, or more precisely a "counit":

e: A -> C

_______

/ \

| |

| |

| |

|\_______/|

| |

| |

| |

| |

\ /

\_____/

And, a wee bit topology shows that these make A into a "coalgebra",

satisfying the "coassociative law" and the left and right "counit

laws". Everything has just been turned upside down!

It's easy to see that the multiplication on A is commutative,

at least for n > 1:

__ __ __ __

/ \ / \ / \ / \

| | | | | | | |

|\__/| |\__/| |\__/| |\__/|

| | | | | | | |

\ \/ / | | | |

\ /` / | | | |

\ / ` / \ \ / /

/ /\ \ \ / /

/ ` / \ \ \_/ /

/ `/ \ \ /

/ /\ \ \ /

| / \ | \ /

| \__/ | = | |

\ / | |

\ / | |

\ / | |

| | | |

| | | |

| ... | | ... |

|' `| |' `|

| | | |

\___/ \___/

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: A tensor A -> C

But now we see this operator comes from a spacetime where two

universes collide and then disappear into nothing:

______ ______

/ \ / \

| | | |

| | | |

| | | |

|\______/| |\______/|

| | | |

| | | |

\ \ / /

\ \ / /

\ \_/ /

\ /

\ /

\ /

\ /

\_________/

To check the Frobenius axiom, we just use associativity:

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

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 |-> 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: C -> A tensor A

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

___________

/ \

/ \

/ \

/ \

/ _ \

/ / \ \

/..... / \ .....\

/ ` / \ ' \

/ `/ \' \

| | | |

| | | |

\ / \ /

\______/ \______/

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!

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

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 tensor 1_A) (1_A tensor m) = m o Delta =

(m tensor 1_A) (1_A tensor Delta)

These equations are a lot more charismatic in pictures! 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 tensor A also

becomes an (A,A)-bimodule, like this:

a (b tensor c) d = ab tensor 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 tensor Y and Y tensor 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:

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:

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: 2Cob -> 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. 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)

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

math.AT/0510664.

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

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. 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.

This generalizes a construction due to Fukuma, Hosono and Kawai,

explained way back in "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

arXiv: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 x n matrices and

defining

g(a,b) = k 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 |-> ab

so for any algebra we can try to define

g(a,b) = tr(L_a L_b)

This bilinear form is nondegenerate precisely when our algebra is

"strongly separable", as defined here:

12) Marcelo Aguiar, A note on strongly separable algebras, available

at http://www.math.tamu.edu/~maguiar/strongly.ps.gz [Broken]

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 x n matrices,

g(a,b) = tr(L_a L_b)

is n times the usual tr(ab). But it's better, in a way. The reason

is that for any strongly separable algebra,

g(a,b) = 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! 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 "separable".

If we use a commutative separable Frobenius algebra to get a 2d TQFT,

it fails to detect handles! That seems sad. But this paper:

13) 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:

Cospan(FinSet) IS THE FREE SYMMETRIC MONOIDAL CATEGORY

ON A COMMUTATIVE SEPARABLE FROBENIUS ALGEBRA.

Here Cospan(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 "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 *separable* commutative Frobenius algebra.

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

First, take the exterior algebra Lambda V 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: Lambda V -> 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 Lambda V 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:

14) Kevin Costello, Topological conformal field theories and

Calabi-Yau categories, available as arxiv: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 all 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!

15) Ingo Runkel, Jens Fjelstad, Jurgen Fuchs, Christoph Schweigert,

Topological and conformal field theory as Frobenius algebras,

available as arXiv: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 get them by 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 modular tensor categories from von

Neumann algebras!

If you want to learn the basics, this book is great - there's a

slightly unpolished version free online:

16) 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:

17) Michael Mueger, Tensor categories: a selective guided tour,

available as arXiv:0804.3587.

Here's a quite different introduction to recent developments, at

least up to 2004:

18) Damien Calaque and Pavel Etingof, Lectures on tensor categories,

available as 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":

19) Hendryk Pfeiffer, Tannaka-Krein reconstruction and a

characterization of modular tensor categories, available as

arXiv:0711.1402.

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

20) Seung-moon Hong, Eric Rowell, Zhenghan Wang, On exotic modular

tensor categories, available as arXiv: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 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 semi-simple 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 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 Mellies 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 separable

Frobenius algebra over the complex numbers looks like

C + C + ... + C + C

It's a direct sum of finitely many copies of C, equipped with its

god-given bilinear form

g(a,b) = 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 separable complex Frobenius algebras as objects;

algebra homomorphisms as morphisms

is equivalent to 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 discuss some nice ways to think about this fact...

and Jamie Vicary even studies how to *categorify* this fact!

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 separable complex Frobenius algebras as objects;

Frobenius homomorphisms as morphisms

is equivalent to the *groupoid* of finite sets. We get FinSet^{op}

if we take algebra homomorphisms, and I guess we get 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:

21) 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.

I explained some of the basics behind this paper in "week174".

But anyway... suppose A is some sort of category with propositions

as objects and proofs as morphisms. So, a morphism

f: a -> b

is a proof that a implies b.

Next, suppose A is a monoidal category and call the tensor product

"or". So, for example, given proofs

f: a -> b, f': a' -> b'

we get a proof

f or f': a or a' -> b or b'

Next, suppose we make the opposite category A^{op} into a monoidal

category, but with a completely different tensor product, that we'll

call "and". And suppose we have a monoidal functor:

not: A -> 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: A^{op} -> 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 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.

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

not

---------->

A A^{op}

<----------

not

be an adjoint equivalence. So, for example, there's a one-to-one

correspondence between proofs

not(a) -> b

and proofs

not(b) -> a

Summarizing: we have a monoidal adjoint equivalence between A

(with one tensor product) and A^{op} (with another). If I'm not

confused, this is just a funny way of talking about a *-autonomous

category. I'm sure someone will correct me if I'm wrong...

...but what matters to me now is that from this data, I can get a

kind of "nondegenerate bilinear form":

g: A x A -> Set

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

not(a) -> b

And here's the punchline: g satisfies the Frobenius axiom in

a categorified way! Namely:

g(a or b, c) is isomorphic to g(a, b or c)

since the first one is the set of proofs

not(a or b) -> c

and the second is the set of proofs

not(a) -> b or c

and by our setup - or just common sense - these are in 1-1

correspondence!

To make this all rigorous and show that *-autonomous categories

are really examples of Frobenius pseudomonoids, we need to use

the machinery of profunctors, also known as distributors. But,

I'm too tired to explain those now! I hope you're tired too...

otherwise I'm not doing my job. You can read more in Street's

paper "Frobenius monoids and pseudomonads".

In short: Frobenius algebras are lurking all over in physics, logic

and quantum logic. There should be some unified explanation of

what's going on! Do you have any ideas?

Finally, here are some books on math and musics that I should read

someday. The first seems more elementary, the second more advanced:

22) 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

23) 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

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Quote of the Week:

'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

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