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This Week's Finds in Mathematical Physics (Week 242)

  1. Dec 20, 2006 #1
    Also available as http://math.ucr.edu/home/baez/week242.html

    December 17, 2006
    This Week's Finds in Mathematical Physics (Week 242)
    John Baez

    This week I'd like to talk about a paper by Jeffrey Morton. Jeff
    is a grad student now working with me on topological quantum field
    theory and higher categories. I've already mentioned his work on
    categorified algebra and quantum mechanics in "week236". He'll be
    finishing his Ph.D. thesis in the spring of 2007 - and as usual,
    that means he's already busy applying for jobs.

    As all you grad students reading this know, applying for jobs is
    pretty scary the first time around: there are some tricks involved,
    and nobody prepares you for it. I remember myself, wondering what
    I'd do if I didn't succeed. Would I have to sell ice cream from one
    of those trucks that plays a little tune as it drives around the
    neighborhood? A job in the financial industry seemed scarcely more
    appealing: less time to think about math, and less ice cream too.

    Luckily things worked out for me... and I'm sure they'll work out
    for Jeff and my other student finishing up this year - Derek Wise,
    who is working on Cartan geometry and MacDowell-Mansouri gravity.

    But, to help them out a bit, I'd like to talk about their work.
    This has been high on my list of interests for the last few years,
    of course, but I've mostly been keeping it under wraps.

    This time I'll talk about Jeff's thesis; next time Derek's. But
    first, let's start with some cool astronomy pictures!

    Here's a photo of Saturn, Saturn's rings, and its moon Dione, taken by
    the Cassini orbiter in October last year:

    1) NASA, Ringside with Dione,

    It's so vivid it seems like a composite fake, but it's not! With
    the Sun shining from below, delicate shadows of the B and C rings
    cover Saturn's northern hemisphere. Dione seems to hover nearby.
    Actually it's 39,000 kilometers away in this photo. It's 1,200
    kilometers in diameter, about the third the size of our Moon.

    Here's a photo of Saturn, its rings, and its moon Mimas, taken
    in November 2004:

    2) NASA, Nature's canvas,

    It's gorgeous, but it takes some work to figure out what's going on!

    The blue stuff in the background is Saturn, with lines created by shadows
    of rings. The bright blue-white stripe near Mimas is sunlight shining
    through a break in the rings called the "Cassini division". The brownish
    stuff near the bottom is the A ring - you can see right through it. Above
    it there's a break and a thinner ring called the F ring. Below it is the
    Cassini division itself.

    This is just one of many photos taken by Cassini and Huyghens, the probe
    that Cassini dropped onto Saturn's moon Titan - see "week210" for more on
    that. You can see more of these photos here:

    3) NASA, Cassini-Huygens, http://saturn.jpl.nasa.gov/

    I hope you see from these beautiful images, and others on This Week's
    Finds, that we are *already in space*. We don't need people up there
    for us to effectively *be there*.

    Alas, not everyone recognizes this. An expensive American program to set
    up a base on the Moon, perhaps as a stepping stone to a manned mission to
    Mars, is starting to drain money from more exciting unmanned missions.
    NASA guesses this program will cost $104 billion up to the time when we
    land on the Moon - again - in 2020. By 2024, the Government Accounting
    Office guesses the price will be $230 billion. By comparison, the
    Cassini-Huygens mission cost just about $3.3 billion.

    And what will be benefits of a Moon base be? It's unclear: at best,
    some vague dream of "space colonization".

    Mind you, I'm in favor of space exploration, and even colonization.
    But, these are very different things!

    Colonies are usually about making money. Governments support them
    in hopes of turning a profit: think Columbus and Isabella, or other
    adventurers funded by colonial powers.

    Right now most of the money lies in near-earth orbit, not on the Moon and
    Mars. Telecommunication satellites and satellite photos are established
    businesses. The next step may be tourism. Dennis Tito, Gregory Olsen and
    Mark Shuttleworth have already paid the Russian government $20 million
    each to visit the International Space Station. This orbits at an altitude
    of about 350 kilometers, in the upper "thermosphere" - the layer of the
    Earth's atmosphere where gases get ionized by solar radiation.

    If this is too pricey for you, wait a few years. Richard Branson's
    company Virgin Galactic plans to give 500 people per year a 7-minute
    experience of weightlessness at a cost of just $200,000 each. Alas,
    you'll only go up 100 kilometers, near the bottom of the thermosphere.

    Some competition may lower the price. Jeff Bezos, the founder of Amazon,
    has bought a lot of land in Texas to a build space port for his company
    Blue Origins. He wants to do test flights by next year, and he eventually
    wants 50 flights a year in a vehicle that holds 3. If you've always looked
    forward to using your seat cushion as a flotation device in the event
    of a water landing, you'll love this:

    "During an abort situation, the crew capsule would separate, using
    small solid-rocket motors to safely recover the space flight participants.
    The abort module containing the solid-rocket motors would then jettison
    from the crew capsule."

    None of this stuff requires any taxpayer funding. It's a bit
    self-indulgent and silly, but it may eventually grow and merge with
    other profit-making forms of space colonization.

    Exploration is a bit different: seeing what's out there, mainly for the
    sake of adventure and understanding. For this we should send machines,
    not people. Machines can be designed to do well in vacuum. People can't -
    not yet. This will probably change when nanotech, AI and cyborg technologies
    kick in. But for now, unmanned probes are the way to go.

    Here are some of the wonderful things we could do, probably all for
    less than setting up a Moon base:

    4) The Laser Interferometer Space Antenna (LISA), http://lisa.jpl.nasa.gov/

    The idea of LISA is to put 3 satellites in a huge equilateral triangle
    following the Earth in its orbit around the Sun, and bounce lasers
    between them to detect gravitational waves (see "week143"). This
    would avoid the ground noise that plagues LIGO (see "week241"), and it
    could detect waves of much lower frequencies. If all works well, it
    could see gravitational waves from the *very* early Universe, long
    before the hot gas enough cooled to let light through. We're talking
    times like 10^{-38} seconds after the Big Bang! That's the biggest
    adventure I can imagine... back to the birth pangs of the Universe.

    Right now LISA is scheduled for launch around 2016. Let's hope it stays
    that way!

    5) Constellation-X, http://constellation.gsfc.nasa.gov/

    This would be a team of X-ray telescopes, combining forces to be 100
    times more powerful than any previous single one. Among many other things,
    Constellation-X could study the X-rays emitted by matter falling into things
    that look like black holes. The redshift of these X-rays is our best test
    of general relativity for very strong gravitational fields. So, it's our
    best way of checking that these black hole candidates really do have event

    In February 2006, when NASA put out their latest budget, they
    said Constellation-X would be "delayed indefinitely".

    6) The Terrestrial Planet Finder (TPF),

    This could study Earth-like planets orbiting stars up to 45 light years
    away. It would consist of two observatories: a visible-light "coronagraph"
    that blocks out the light from a star so it can see nearby fainter objects,
    and an infrared interferometer made of several units flying in formation.

    In February 2006, NASA halted work on the TPF.

    7) The Nuclear Spectroscopic Telescope Array (NuStar),

    This is an orbiting observatory with three telescopes, designed to see
    hard X-rays. It could conduct a thorough survey of black hole candidates
    throughout the universe, it could study relativistic jets of particles
    from the cores of active galaxies (which are probably also black holes),
    and it could study young supernova remnants - hot new neutron stars.

    NASA suddenly canceled work on NuStar in February 2006.

    8) Dawn, http://dawn.jpl.nasa.gov/

    The Dawn mission seeks to understand the early Solar System by probing
    the asteroid belt and taking a good look at Ceres and Vesta. Ceres is
    the largest asteroid of all, 950 kilometers in diameter. It seems
    have a rocky core, a thick mantle of water ice, and a thin dusty outer
    crust. Vesta is the second largest, about 530 kilometers in diameter.
    It's very different from Ceres: it's not round, and it's all rock. A
    certain group of stony meteorites called "HED meteorites" are believed
    to be pieces of Vesta!

    NASA cancelled the Dawn mission in March 2006 - but later that month,
    they changed their minds.

    It's depressing to contemplate all the wonderful things we could miss
    while spending hundreds of billions to "send canned primates to Mars",
    as Charles Stross so cleverly put it in his novel Accelerando (see
    "week222"). I'm all for humanity spreading through space. I just don't
    think we should do it in a clunky, low-tech way like setting up a base
    on the Moon where astronauts sit around and... what, play golf? It's
    like something out of old science fiction!

    To cheer myself up again, here's a picture of the sun:

    9) Joanne Hewett, Sun Shots, http://cosmicvariance.com/2006/10/13/sun-shots/

    It was taken not with light, but with *neutrinos*. It was made at
    the big neutrino observatory in Japan, called Super-Kamiokande. It
    took about 504 days and nights to make.

    That's right - nights! Neutrinos go right through the Earth.

    As you probably know, neutrinos oscillate between three different
    kinds, but only electron neutrinos are easy to detect, so we see about
    third as many neutrinos from the Sun as naively expected. That's
    the kind of thing they're studying at Super-Kamiokande.

    But what I want to know is: what's the glare" in this picture?
    Neutrinos are made by the process of fusion, which involves this

    proton + electron -> neutron + electron neutrino

    Fusion mostly happens in the Sun's core, which has a density of 160
    grams per cubic centimeter (10 times denser than lead) and a temperature
    of 15 million kelvin (300 thousand times hotter than the "broil" setting
    on an American oven).

    So, what's the disk in this picture: the whole Sun, or the Sun's core?
    And what's the glare?

    Okay, now for some serious mathematical physics:

    10) Jeffrey Morton, A double bicategory of cobordisms with corners,
    available as math.CT/0611930.

    People have been talking a long time about topological quantum field
    theory and higher categories. The idea is that categories, 2-categories,
    3-categories and the like can describe how manifolds can be chopped into
    little pieces - or more precisely, how these little pieces can be glued
    together to form manifolds. Then the problem of doing quantum field
    theory on some manifold can be reduced to the problem of doing it on
    these pieces and gluing the results together. This works easiest if
    the theory is "topological", not requiring a background metric.

    There's a lot of evidence that this is a good idea, but getting the details
    straight has proved tough, even at the 2-category level. This is what
    Morton does, in a rather clever way. Very roughly, his idea is to use
    something I'll call a "weak double category", and prove that these:

    (n-2)-dimensional manifolds

    (n-1)-dimensional manifolds with boundary

    n-dimensional manifolds with corners

    give a weak double category called nCob_2. The proof is a cool mix of
    topology and higher category theory. He then shows that this particular
    weak double category can be reinterpreted as something a bit more
    familiar - a "weak 2-category".

    In the rest of his thesis, Jeff will use this formalism to construct
    some examples "extended TQFTs", which are roughly maps of weak 2-categories

    Z: nCob_2 -> 2Vect

    where 2Vect is the weak double category of "2-vector spaces". He's
    focusing on some extended TQFTs called the Dijkgraaf-Witten models,
    coming from finite groups.

    But, he's also thought about the case where the finite group is
    replaced by the Lie group SU(2). In this case we get something a
    lot like an extended TQFT, but not quite, called the Ponzano-Regge
    model. In this case of 3d spacetime, this is a nice theory of
    quantum gravity. And, as I hinted back in "week232", we can let 2d
    space in this model be a manifold with *boundary* by poking little
    holes in space - and these holes wind up acting like particles!

    So, we get a relation like this:

    (n-2)-dimensional manifolds MATTER

    (n-1)-dimensional manifolds with boundary SPACE

    n-dimensional manifolds with corners SPACETIME

    which is really quite cool.

    It would be fun to talk about this. However, to understand Morton's
    work more deeply, you need to understand a bit about "weak double
    categories". He explains them quite nicely, but I think I'll spend
    the rest of this Week's Finds giving a less detailed introduction,
    just to get you warmed up.

    This chart should help:


    LAWS HOLDING strict strict
    AS EQUATIONS 2-categories double categories

    LAWS HOLDING weak weak
    UP TO ISOMORPHISM 2-categories double categories

    2-categories are good for describing how to glue together 2-dimensional
    things that, at least in some abstract sense, are shaped like *bigons*.
    A "bigon" is a disc with its boundary divided into two halves. Here's
    my feeble ASCCI rendition of a bigon:

    / \
    / || \
    X o ||B o Y
    \ \/ /
    \ /

    The big arrow indicates that we think of the bigon B as "going from"
    the top semicircle, f, to the bottom semicircle, g. Similarly, we
    think of the arcs f and g as going from the point X to the point Y.

    Similarly, double categories are good for describing how to glue together
    2-dimensional gadgets that are shaped like *squares*:

    X o---->----o X'
    | |
    g v S v g'
    | |
    Y o---->----o Y'

    Both 2-categories and double categories come in "strict" and "weak"
    versions. The strict versions have operations satisfying a bunch of
    laws "on the nose", as equations. In the weak versions, these laws
    hold up to isomorphism whenever possible.

    A few more details might help....

    A 2-category has a set of objects, a set of morphisms f: X -> Y going
    from any object X to to any object Y, and a set of 2-morphisms T: f => g
    going from any morphism f: X -> Y to any morphism g: X -> Y. We can
    visualize the objects as dots:


    the morphisms as arrows:

    X o---->----o Y

    and the 2-morphisms as bigons:

    / \
    / || \
    X o ||B o Y
    \ \/ /
    \ /

    We can compose morphisms like this:

    f g fg
    o---->----o---->----o gives o--->---o
    X Y Z X Y

    We can also compose 2-morphisms vertically:

    f f
    ---->---- --->---
    / S \ / \
    / g \ / \
    X o ----->----- o Y gives X o ST o Y
    \ T / \ /
    \ / \ /
    ---->---- --->---
    h h

    and horizontally:

    f f' ff'
    --->--- --->--- --->---
    / \ / \ / \
    / \ / \ / \
    X o S o T o Z gives X o S.T o Y
    \ / \ / \ /
    \ / \ / \ /
    --->--- --->--- --->---
    g g' gg'

    There are also a bunch of laws that need to hold. I don't want to
    list them; you can find them in Jeff's paper (also see "week80").
    I just want to emphasize how a strict 2-category is different from
    a weak one.

    In a strict 2-category, the composition of morphisms is associative
    on the nose:

    (fg)h = f(gh)

    and there are identity morphisms that satisfy these laws on the nose:

    1f = f = f1

    In a weak 2-category, these equations are replaced by 2-isomorphisms - that
    is, invertible 2-morphisms. And, these 2-isomorphisms need to satisfy new
    equations of their own!

    What about double categories?

    Double categories are like 2-categories, but instead of bigons, we have

    More precisely, a double category has a set of objects:


    a set of horizontal arrows:

    X o---->----o X'

    a set of vertical arrows:

    X o
    g v
    Y o

    and a set of squares:

    X o---->----o X'
    | |
    g v S v g'
    | |
    Y o---->----o Y'

    We can compose the horizontal arrows like this:

    f f' f.f'
    o---->----o---->----o gives o--->---o
    X Y Z X Y

    We can compose the vertical arrows like this:

    X o
    g v o
    | |
    Y o gives gg' v
    | |
    g' v o
    Y o

    And, we can compose the squares both vertically:

    X o---->----o X'
    | | f
    g v S v g' X o---->----o X'
    | | | |
    Y o---->----o Y' gives gh v SS' v g'h'
    | | | |
    h v S' v h' Z o---->----o Z'
    | | f'
    Z o---->----o Z'

    and horizontally:

    f Y g f.g
    X o---->----o---->----o Z X o---->----o Z
    | | | | |
    h v S v S' v h' h v S.S' v h'
    | | | | |
    X' o---->----o---->----o Z' X' o---->----o Z'
    f' Y' g' f'.g'

    In a strict double category, both vertical and horizontal composition
    of morphisms is associative on the nose:

    (fg)h = f(gh) (f.g).h = f.(g.h)

    and there are identity morphisms for both vertical and horizontal
    composition, which satisfy the usual identity laws on the nose.

    In a weak double category, we want these laws to hold only up to
    isomorphism. But, it turns out that this requires us to introduce
    bigons as well! The reason is fascinating but too subtle to explain
    here. I didn't understand it until Jeff pointed it out. But, it
    turns out that Dominic Verity had already introduced the right concept
    of weak double category - a gadget with both squares and bigons - in
    *his* Ph.D. thesis a while back:

    11) Dominic Verity, Enriched categories, internal categories, and
    change of base, Ph.D. dissertation, University of Cambridge, 1992.

    Interestingly, if you weaken *only* the laws for vertical composition,
    you don't need to introduce bigons. The resulting concept of
    "horizontally weak double category" has been studied by Grandis and Pare:

    12) Marco Grandis and Bob Pare, Limits in double categories, Cah.
    Top. Geom. Diff. Cat. 40 (1999), 162-220.

    Marco Grandis and Bob Pare, Adjoints for double categories, Cah.
    Top. Geom. Diff. Cat. 45 (2004), 193-240. Also available at

    and more recently by Martin Hyland's student Richard Garner:

    13) Richard Garner, Double clubs, available as math.CT/0606733

    and Tom Fiore:

    14) Thomas M. Fiore, Pseudo algebras and pseudo double categories,
    available as math.CT./0608760.

    At this point I should admit that the terminology in this whole
    field is a bit of a mess. I've made up simplified terminology
    for the purposes of this article, but now I should explain how it
    maps to the terminology most people use:


    strict 2-category 2-category
    weak 2-category bicategory
    strict double category double category
    weak double category double bicategory
    horizontally weak double category pseudo double category

    Verity used the term "double bicategory" to hint that his gadgets
    have both squares and bigons, so they're like a blend of double
    categories and bicategories. It's a slightly unfortunate term, since
    experts know that a double category is a category object in Cat, but
    Verity's double bicategories are not bicategory objects in BiCat.
    Morton mainly uses Verity's double bicategories - but in the proof of
    his big theorem, he also uses bicategory objects in BiCat.

    There's a lot more to say, but I'll stop here and let you read the
    rest in Jeff's paper!


    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


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


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


    If you just want the latest issue, go to

  2. jcsd
  3. Dec 24, 2006 #2
    John Baez wrote:
    > To cheer myself up again, here's a picture of the sun:
    > 9) Joanne Hewett, Sun Shots, http://cosmicvariance.com/2006/10/13/sun-shots/
    > It was taken not with light, but with *neutrinos*.
    > But what I want to know is: what's the glare" in this picture?
    > So, what's the disk in this picture: the whole Sun, or the Sun's core?
    > And what's the glare?

    A recurring theme in experimental physics is errorbars and an analysis
    of experimental resolutions; that is important here. As the discussion
    in that link explains, the visible sun is smaller than a single pixel of
    the picture. That means that all of the "structure" and "glare" are
    artifacts of the instrumentation. The detector simply does not have good
    enough angular resolution to see details -- it is a tour de force to
    even get within a few tens of degrees of the sun!

    Tom Roberts
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