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

  1. Aug 8, 2008 #1
    Also available at http://math.ucr.edu/home/baez/week268.html

    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

    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

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

    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

    -----------------------------------------------------------------------
    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at

    http://math.ucr.edu/home/baez/

    For a table of contents of all the issues of This Week's Finds, try

    http://math.ucr.edu/home/baez/twfcontents.html

    A simple jumping-off point to the old issues is available at

    http://math.ucr.edu/home/baez/twfshort.html

    If you just want the latest issue, go to

    http://math.ucr.edu/home/baez/this.week.html
     
  2. jcsd
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