June 20, 2008

This Week's Finds in Mathematical Physics (Week 266)

John Baez

I'm at this workshop now, and I want to talk about it:

1) Workshop on Categorical Groups, June 16-20, 2008, Universitat de Barcelona, organized by Pilar Carrasco, Josep Elgueta, Joachim Kock and Antonio Rodríguez Garzón, http://mat.uab.cat/~kock/crm/hocat/cat-groups/

But first, the moon of the week - and a bit about that mysterious fellow Pythagoras, and the Pythagorean tuning system.

Here's a picture of Jupiter's moon Io:

1) Io in True Color, Astronomy Picture of the Day, http://antwrp.gsfc.nasa.gov/apod/ap040502.html

It's yellow! - a world covered with sulfur spewed from volcanos, burning hot inside from intense tidal interactions with Jupiter's mighty gravitational field... but frigid at the surface.

Last week I talked about something called the "Pythagorean pentagram". That's a cool name - but it's far from clear who first discovered this entity, so I started feeling a bit guilty for using it, and I started wondering what we actually know about Pythagoras or the mathematical vegetarian cult he supposedly launched. Tim Silverman pointed me to a scholarly book on the subject:

2) Walter Burkert, Lore and Science in Ancient Pythagoreanism, Harvard U. Press, Cambridge, Massachusetts, 1972.

It turns out we know very little about Pythagoras: a few grains of solid fact, surrounded by a huge cloud of stories that grows larger and larger as we move further and further away from the 6th century BC, when he lived. This is especially true when it comes to his contributions to mathematics. The infamous pseudohistorian Eric Temple Bell begins his book "The Magic of Numbers" as follows:

The hero of our story is Pythagoras. Born to immortality five hundred years before the Christian era began, this titanic spirit overshadows western civilization. In some respects he is more vividly alive today than he was in his mortal prime twenty-five centuries ago, when he deflected the momentum of prescientific history toward our own unimagined scientific and technological culture. Mystic, philosopher, experimental physicist, and mathematician of the first rank, Pythagoras dominated the thought of his age and foreshadowed the scientifi mysticisms of our own.
But, there's no solid evidence for any of this, except perhaps his interest in mysticism and numerology and the incredible growth of his legend as the centuries pass. We're not even sure he proved the "Pythagorean theorem", much less all the other feats that have been attributed to him. As Burkert explains:
No other branch of history offers such temptations to conjectural reconstruction as does the history of mathematics. In mathematics, every detail has its fixed and unalterable place in a nexus of relations, so that it is often possible, on the basis of a brief and casual remark, to reconstruct a complicated theory. It is not surprising, then, that gap in the history of mathematics that was opened up by a critical study of the evidence about Pythagoras has been filled by a whole succession of conjectural supplements.
There's a new book out on Pythagoras:

3) Kitty Ferguson, The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space, Walker and Company, 2008.

The subtitle is sensationalistic, exactly the sort of thing that would make Burkert cringe. But the book is pretty good, and Ferguson is honest about this: after asking "What do we know about Pythagoras?", she lists everything we know in one short paragraph, and then emphasizes: that's all.

He was born on the island of Samos sometime around 575 BC. He went to Croton, a city in what is now southern Italy. He died around 495 BC. We know a bit more - but not much.

It's much easier to learn about the Renaissance "neo-Pythagoreans". This book is a lot of fun, though too romantic to be truly scholarly:

4) S. K. Heninger, Jr., Touches of Sweet Harmony: Pythagorean Cosmology and Renaissance Poetics, The Huntington Library, San Marino, California, 1974.

It seems clear that the Renaissance neo-Pythagoreans, and even the Greek Pythagoreans, and perhaps even old Pythagoras himself were much taken with something called the tetractys:


        o      o
     o      o     o

  o     o      o     o

To appreciate the tetractys, you have to temporarily throw out modern scientific thinking and get yourself in the mood of magical thinking - or "correlative cosmology", which tries to understand the universe by setting up elaborate correspondences between this, that, and the other thing. To the Pythagoreans, the four rows of the tetractys represented the point, line, triangle and tetrahedron. But the "fourness" of the tetractys also represented the four classical elements: earth, air, water and fire. It's fun to compare these early groping attempts to impose order on the universe to later, less intuitive but far more predictively powerful schemes like the Periodic Table or the Standard Model. So, let's take a look!

The Renaissance thinkers liked to organize the four elements using a chain of analogies running from light to heavy:

fire : air :: air : water :: water : earth

Them also organized them in a diamond, like this:

         hot           dry
    AIR                    EARTH
         wet          cold

Sometimes they even put a fifth element in the middle: the "quintessence", or "aether", from which heavenly bodies were made. And following Plato's Timaeus dialog, they set up an analogy like this:

fire         tetrahedron 
air          octahedron
water        icosahedron 
earth        cube
quintessence dodecahedron
This is cute! Fire feels pointy and sharp like tetrahedra, while water rolls like round icosahedra, and earth packs solidly like cubes. Dodecahedra are different than all the rest, made of pentagons, just as you might expect of "quintessence". And air... well, I've never figured out what air has to do with octahedra. You win some, you lose some - and in correlative cosmology, a discrepancy here and there doesn't falsify your ideas.

The tetractys also took the Pythagoreans in other strange directions. For example, who said this?

"What you suppose is four is really ten..."

A modern-day string theorist talking to Lee Smolin about the dimension of spacetime? No! Around 150 AD, the rhetorician Lucian of Samosata attributed this quote to Pythagoras, referring to the tetratkys and the fact that it has 1 + 2 + 3 + 4 = 10 dots. This somehow led the Pythagoreans to think the number 10 represented "perfection". If there turn out to be 4 visible dimensions of spacetime together with 6 curled-up ones explaining the gauge group U(1) × SU(2) × SU(3), maybe they were right.

Pythagorean music theory is a bit more comprehensible: along with astronomy, music is one of the first places where mathematical physics made serious progress. The Greeks, and the Babylonians before them, knew that nice-sounding intervals in music correspond to simple rational numbers. For example, they knew that the octave corresponds to a ratio of 2:1. We'd now call this a ratio of frequencies; one can get into some interesting scholarly arguments about when and how well the Greeks knew that sound was a vibration, but never mind - read Burkert's book if you're interested.

Whatever these ratios meant, the Greeks also knew that a fifth corresponds to a ratio of 3:2, and a fourth to 4:3.

By the way, if you don't know about musical intervals like "fourths" and "fifths", don't feel bad. I won't explain them now, but you can learn about them and hear them here:

5) Brian Capleton, Musical intervals, http://www.amarilli.co.uk/music/intervs.htm

and then practice recognizing them:

6) Ricci Adams, Interval ear trainer, http://www.musictheory.net/trainers/html/id90_en.html

If you nose around Capleton's website, you'll see he's quite a Pythagorean mystic himself!

Anyway, at some moment, lost in history by now, people figured out that the octave could be divided into a fourth and a fifth:

2/1 = 4/3 × 3/2

And later, I suppose, they defined a whole tone to be the difference, or really ratio, between a fifth and a fourth:

(3/2)/(4/3) = 9/8

So, when you go up one whole tone in the Pythagorean tuning system, the higher note should vibrate 9/8 as fast as the lower one. If you try this on a modern keyboard, it looks like after going up 6 whole tones you've gone up an octave. But in fact if you buy the Pythagorean definition of whole tone, 6 whole tones equals

(9/8)6 = 531441 / 262144 ≅ 2.027286530...

which is, umm, not quite 2!

Another way to put it is that if you go up 12 fifths, you've almost gone up 7 octaves, but not quite: the so-called circle of fifths doesn't quite close, since

(3/2)12 / 27 = 531441 / 524288 ≅ 1.01264326...

This annoying little discrepancy is called the "Pythagorean comma".

This sort of discrepancy is an unavoidable fact of mathematics. Our ear likes to hear frequency ratios that are nice simple rational numbers, and we'd also like a scale where the notes are evenly spaced - but we can't have both. Why? Because you can't divide an octave into equal parts that are rational ratios of frequencies. Why? Because a nontrivial nth root of 2 can never be rational.

So, irrational numbers are lurking in any attempt to create an equally spaced (or as they say, "equal-tempered") tuning system.

You might imagine this pushed the Pythagoreans to confront irrational numbers. This case has been made by the classicist Tannery, but Burkert doesn't believe it: there's no written evidence suggesting it.

You could say the existence of irrational numbers is the root of all evil in music. Indeed, the diminished fifth in an equal tempered scale is called the "diabolus in musica", or "devil in music", and it has a frequency ratio equal to the square root of 2.

Or, you could say that this built-in conflict is the spice of life! It makes it impossible for harmony to be perfect and therefore dull.

Anyway, Pythagorean tuning is not equal-tempered: it's based on making lots of fifths equal to exactly 3/2. So, all the frequency ratios are fractions built from the numbers 2 and 3. But, some of them are nicer than others:

first = 1/1
second = 9/8
third = 81/64
fourth = 4/3
fifth = 3/2
sixth = 27/16
seventh = 243/128
octave = 2/1

As you can see, the third, sixth and seventh are not very nice: they're complicated fractions, so they don't sound great. They're all a bit sharp compared to the following tuning system, which is a form of "just intonation":

first = 1/1                          
second = 9/8                        
third = 5/4                           
fourth = 4/3                           
fifth = 3/2
sixth = 5/3
seventh = 15/8
octave = 2/1

Just intonation brings in fractions involving the number 5, which we might call the "quintessence" of music: we need it to get a nice-sounding third. A long and interesting tale could be told about this tuning system - but not now. Instead, let's just see how the third, sixth and seventh differ:

Here you can learn more about Pythagorean tuning, and hear it in action:

7) Margo Schulter, Pythagorean tuning and medieval polyphony, http://www.medieval.org/emfaq/harmony/pyth.html

8) Reginald Bain, A Pythagorean tuning of the diatonic scale, http://www.music.sc.edu/fs/bain/atmi02/pst/index.html

There's also a murky relation between Pythagorean tuning and something called the "Platonic Lambda". This is a certain way of labelling the edges of the tetractys by powers of 2 on one side, and powers of 3 on the other:


        2     3
     4           9

  8                 27

I can't help wanting to flesh it out like this, so going down and to the left is multiplication by 2, while going down and to the right is multiplication by 3:


        2      3
     4      6     9

  8    12     18     27

So, I was pleased when in Heninger's book I saw the numbers on the bottom row in a plate from a 1563 edition of "De Natura Rerum", a commentary on Plato's Timaeus written by the Venerable Bede sometime around 700 AD!

In this plate, the elements fire, air, water and earth are labelled by the numbers 8, 12, 18 and 27. This makes the aforementioned analogies:

fire : air :: air : water :: water : earth

into strict mathematical proportions:

8 : 12 :: 12 : 18 :: 18 : 27

Cute! Of course it doesn't do much to help us understand fire, air, earth and water. But, it goes to show how people have been struggling a long time to find mathematical patterns in nature. Most of these attempts don't work. Occasionally we get lucky... and over the millennia, these scraps of luck added up to the impressive theories we have today.

Next: the categorical groups workshop here in Barcelona!

A "categorical group", also called a "2-group", is a category that's been equipped with structures mimicking those of a group: a product, identity, and inverses, satisfying the usual laws either "strictly" as equations or "weakly" as natural isomorphisms. Pretty much anything people do with groups can also be done with 2-groups. That's a lot of stuff - so there's a lot of scope for exploration! There's a powerful group of algebraists in Spain engaged in this exploration, so it makes sense to have this workshop here.

Let me say a little about some of the talks we've had so far. I'll mainly give links, instead of explaining stuff in detail.

On Monday, I kicked off the proceedings with this talk:

9) John Baez, Classifying spaces for topological 2-groups, http://math.ucr.edu/home/baez/barcelona/

Just as we can try to classify principal bundles over some space with any fixed group as gauge group, we can try to classify "principal 2-bundles" with a given "gauge 2-group". It's a famous old theorem that for any topological group G, we can find a space BG such that principal G-bundles over any mildly nice space X are classified by maps from X to BG. (Homotopic maps correspond to isomorphic bundles.) A similar result holds for topological 2-groups!

Indeed, Baas Bökstedt and Kro did something much more general for topological 2-categories:

10) Nils Baas, Marcel Bökstedt and Tore Kro, 2-Categorical K-theories, available as arXiv:math/0612549.

Just as a group is a category with one object and with all morphisms being invertible, a 2-group is a 2-category with one object and all morphisms and 2-morphisms invertible. But the 2-group case is worthy of some special extra attention, so Danny Stevenson studied that with a little help from me:

11) John Baez and Danny Stevenson, The classifying space of a topological 2-group, available as arXiv/0801.3843

and that's what I talked about. If you're also interested in classifying spaces of 2-categories that aren't topological, just "discrete", you should try these:

12) John Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, available at http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html

13) Manuel Bullejos and A. Cegarra, On the geometry of 2-categories and their classifying spaces, available at http://www.ugr.es/%7Ebullejos/geometryampl.pdf

14) Manuel Bullejos, Emilio Faro and Victor Blanco, A full and faithful nerve for 2-categories, Applied Categorical Structures 13 (2005), 223-233. Also available as arXiv:math/0406615.

On Monday afternoon, Bruce Bartlett spoke on a geometric way to understand representations and "2-representations" of ordinary finite groups. You can see his talk here, and also a version which has less material, explained in a more elementary way:

15) Bruce Bartlett, The geometry of unitary 2-representations of finite groups and their 2-characters, talk at the Categorical Groups workshop in Barcelona, June 16, 2008, available at http://brucebartlett.postgrad.shef.ac.uk/research/Barcelona.pdf

Bruce Bartlett, The geometry of 2-representations of finite groups, talk at the Max Kelly Conference, Cape Town, 2008, available at http://brucebartlett.postgrad.shef.ac.uk/research/MaxKellyTalk.pdf

Both talks are based on this paper:

16) Bruce Bartlett, The geometry of unitary 2-representations of finite groups and their 2-characters, draft available at http://brucebartlett.postgrad.shef.ac.uk/research/Max%20Kelly%20Proceedings.pdf

The first big idea here is that the category of representations of a finite group G is equivalent to some category where an object X is a complex manifold on which G acts, equipped with an invariant hermitian metric and an equivariant U(1) bundle. A morphism from X to Y in this category is not just the obvious sort of map; instead, it's diagram of maps shaped like this:

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

This is called a "span". So, we're seeing a very nice extension of the Tale of Groupoidification, which began in "week247" and continued up to "week257", when it jumped over to my seminar.

But Bruce doesn't stop here! He then categorifies this whole story, replacing representations of G on Hilbert spaces by representations on 2-Hilbert spaces, and replacing U(1) bundles by U(1) gerbes. This is quite impressive, with nice applications to a topological quantum field theory called the Dijkgraaf-Witten model.

Next, to handle the TQFT called Chern-Simons theory, Bruce plans to replace the finite group G by a compact Lie group. Another, stranger direction he could go is to replace G by a finite 2-group. Then he'd make contact with the categorified Dijkgraaf-Witten TQFT studied in these papers:

17) David Yetter, TQFT's from homotopy 2-types, Journal of Knot Theory and its Ramifications 2 (1993), 113-123.

18) Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, I: Formal maps and crossed C-algebras, available as arXiv:math/0512032.

Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, II: Simplicial formal maps, in Categories in Algebra, Geometry and Mathematical Physics, eds. A. Davydov et al, Contemp. Math 431, AMS, Providence Rhode Island, 2007, 375-403. Also available as arXiv:math/0512034.

19) João Faria Martins and Timothy Porter, On Yetter's invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups, avilable as arXiv:math/0608484.

As the last paper explains, we can also think of this TQFT as a field theory where the "field" on a spacetime X is a map

f: X → BG

where BG is the classifying space of the 2-group G.

Given all this, it's natural to contemplate a further generalization of Bruce's work where G is a Lie 2-group. Unfortunately, Lie 2-groups don't have many representations on 2-Hilbert space of the sort I've secretly been talking about so far: that is, finite-dimensional ones.

So we may, perhaps, need to ponder representations of Lie 2-groups on infinite-dimensional 2-Hilbert spaces.

Luckily, that's just what Derek Wise spoke about on Wednesday morning! His talk also included some pictures with intriguing relations to the pictures in Bruce's talk. You can see the slides here:

19) Derek Wise, Representations of 2-groups on higher Hilbert spaces, http://math.ucdavis.edu/~derek/talks/barcelona2008.pdf

They make a nice introduction to a paper he's writing with Aristide Baratin, Laurent Freidel and myself. Our work uses ideas like measurable fields of Hilbert spaces, which are already important for understanding infinite-dimensional unitary group representations. But if you're less fond of analysis, jump straight to pages 20, 23 and 25, where he gives a geometrical interpretation of these infinite-dimensional representations, along with the intertwining operators between them... and the "2-intertwining operators" between those.

This work relies heavily on the work of Crane, Sheppeard and Yetter, cited in "week210" - so check out that, too!

There's much more to say, but I'm running out of steam, so I'll just mention a few more talks: Enrico Vitale's talk on categorified homological algebra, and the talks by David Roberts and Aurora del Río on the fundamental 2-group of a topological space.

To set these in their proper perspective, it's good to recall the periodic table of n-categories, mentioned in "week49":

                   k-tuply monoidal n-categories 

              n = 0           n = 1             n = 2

k = 0         sets          categories         2-categories

k = 1        monoids         monoidal           monoidal
                            categories        2-categories

k = 2       commutative      braided            braided
             monoids         monoidal           monoidal
                            categories        2-categories 

k = 3         " "           symmetric           sylleptic
                             monoidal           monoidal 
                            categories        2-categories  

k = 4         " "             " "               symmetric

k = 5         " "             " "                "  "

The idea here is that an (n+k)-category with only one j-morphism for j < k acts like an n-category with extra bells and whistles: a "k-tuply monoidal n-category". This idea has not been fully established, and there are some problems with naive formulations of it, but it's bound to be right when properly understood, and it's useful for anyone trying to understand the big picture of mathematics.

Now, an n-category with everything invertible is called an "n-groupoid". Such a thing is believed to be essentially the same as a "homotopy n-type", meaning a nice space, like a CW complex, with vanishing homotopy groups above the nth - where we count homotopy equivalent spaces as the same. If we accept this, the n-groupoid version of the Periodic Table can be understood using homotopy theory. It looks like this:

                   k-tuply groupal n-groupoids 

              n = 0           n = 1             n = 2

k = 0         sets           groupoids        2-groupoids

k = 1        groups          2-groups          3-groups  
k = 2       abelian          braided           braided
            groups           2-groups          3-groups

k = 3         " "            symmetric         sylleptic
                             2-groups          3-groups 

k = 4         " "              " "             symmetric

k = 5         " "              " "               " "

Most of this workshop has focused on 2-groups. But abelian groups are especially interesting and nice, and there's a huge branch of math called "homological algebra" that studies categories similar to the category of abelian groups. These are called "abelian categories". In an abelian category, you've got direct sums, kernels, cokernels, exact sequences, chain complexes and so on - all things you're used to in the category of abelian groups!

Can we categorify all this stuff? Yes - and that's what Enrico Vitale is busy doing! He started by telling us how all these ideas generalize from abelian groups to symmetric 2-groups, and how they change.

For example, besides the "kernel" and "cokernel", we also need extra concepts. The reason is that the kernel of a homomorphism says if the homomorphism is one-to-one, while its cokernel says if it's onto. Functions can be nice in two basic ways: they can be one-to-one, or onto. But because categories have an extra level, functors between them can be nice in three ways, called "faithful", "full" and "essentially surjective". So, we need more than just the kernel and cokernel to say what's going on. We also need the "pip" and "copip".

The concepts of exact sequence and chain complex get subtler, too. You can read about these things here:

20) Aurora del Río, Martínez-Moreno and Enrico Vitale, Chain complexes of symmetric categorical groups, JPAA 196 (2005), 279-312. Also available at http://www.math.ucl.ac.be/membres/vitale/SCG-compl3.pdf

21) Pilar Carrasco, Antonio Garzón and Enrico Vitale, On categorical crossed modules, TAC 16 (2006), 85-618, available as http://tac.mta.ca/tac/volumes/16/22/16-22abs.html

By generalizing properties of the category of abelian groups, people invented the concept of "abelian category". Similarly, Vitale told us a definition of "2-abelian 2-category", obtained by generalizing properties of the 2-category of symmetric 2-groups. I believe this is discussed here:

22) Mathieu Dupont: Catégories abéliennes en dimension 2, Ph.D. Thesis, Université Catholique de Louvain, 2008. Available in English as arXiv:0809.1760. Original available at http://hdl.handle.net/2078.1/12735

Mathieu Dupont is defending his dissertation on June 30th. I hope he puts it on the arXiv after that. (He did!)

All this stuff gets even more elaborate as we move to n-groups for higher n. To some extent this is the subject of homotopy theory, but one also wants a more explicitly algebraic approach. See for example:

23) Giuseppe Metere: The ziqqurath of exact sequences of n-groupoids, Ph.D. Thesis, Università di Milano, 2008. Also available at arXiv:0802.0800.

The relation between 2-groups and topology is made explicit using the concept of "fundamental 2-group". Just as every space equipped with a basepoint has a fundamental group, it has a fundamental 2-group. And for a homotopy 2-type, this 2-group captures everything about the space - at least if we count homotopy equivalent spaces as the same.

David Roberts prepared an excellent talk about the fundamental 2-group of a space for this workshop. Unfortunately, he was unable to come. Luckily, you can still see his talk:

24) David Roberts, Fundamental 2-groups and 2-covering spaces, http://golem.ph.utexas.edu/category/2008/06/fundamental_2groups_and_2cover.html

The basic principle of Galois theory says that covering spaces of a connected space are classified by subgroups of its fundamental group. Here Roberts explains how "2-covering spaces" of a connected space are classified by "sub-2-groups" of its fundamental 2-group!

Aurora del Río spoke on fundamental 2-groups and their application to K-theory. Whenever we have a fibration of pointed spaces

F → E → B

we get a long exact sequence of homotopy groups

... → πn(F) → πn(E) → πn(B) → πn-1(F) → ...

This is a standard tool in algebraic topology; I sketched how it works in "week151".

Now, the nth homotopy group of a space X, written πn(X), is just the fundamental group of the (n-1)-fold loop space of X. So, the Spanish categorical group experts define the nth "homotopy 2-group" of a space X to be the fundamental 2-group of an iterated loop space of X. And, it turns out that any fibration of spaces gives a long exact sequence of homotopy 2-groups!

I was surprised by this, but in retrospect I shouldn't have been. Any fibration gives a "long exact sequence of iterated loop spaces":

... → LnF → Ln E → Ln B → Ln-1F → ...

So, as soon as we have a definition of "fundamental n-groupoids" and long exact sequences of n-groupoids, and can show that taking the fundamental n-groupoid preserves exactness, we can get a long exact sequence of fundamental n-groupoids. If we simply define a fundamental n-groupoid to be a homotopy n-type, this should not be hard.

But this was just the warmup for Aurora's talk, which was about K-theory. Quillen set up modern algebraic K-theory by defining the K-groups of a ring R to be the homotopy groups of a certain space called BGL(R)+. In here talk, Aurora defined the K-2-groups of a ring in the same way, but using homotopy 2-groups! And then she went ahead and studied them...

The slides for Aurora's talk are - as for many of the talks - available from the workshop's website:

25) Aurora del Río, Algebraic K-theory for categorical groups, http://mat.uab.cat/~kock/crm/hocat/cat-groups/slides/delRio.pdf

See also the paper she and Antonio Garzón wrote on this topic:

26) Antonio Garzón and Aurora del Río, On algebraic K-theory categorical groups, http://www.ugr.es/~agarzon/K-thCG.pdf

Also try these other papers:

26) Antonio Garzón and Aurora del Río, Low-dimensional cohomology of categorical groups, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 (2003), 247-280. Available at http://www.numdam.org/numdam-bin/fitem?id=CTGDC_2003__44_4_247_0

This one gets into K-theory:

27) Antonio Garzón and Aurora del Río, The Whitehead categorical group of derivations, Georgian Mathematical Journal 09 (2002), 709-721. Available at http://www.heldermann.de/GMJ/GMJ09/GMJ094/gmj09053.htm

Addenda: Writing the above stuff caused me to miss Behrang Noohi's talk on using diagrams called "butterflies" to efficiently describe weak homomorphisms between strict 2-groups (in the guise of crossed modules). Luckily Tim Porter summarized it at the n-Category Café:

27) Timothy Porter, Behrang Noohi on butterflies and weak morphisms between 2-groups, available at http://golem.ph.utexas.edu/category/2008/06/behrang_noohi_on_butterflies_a.html

For more details, you can't beat the original paper:

28) Behrang Noohi, On weak maps between 2-groups, available as arXiv:math/0506313.

Also at the n-Category Café, Bruce Bartlett discussed Tim Porter's talk at the categorical groups workshop:

29) Bruce Bartlett, Tim Porter on formal homotopy quantum field theories and 2-groups, available at http://golem.ph.utexas.edu/category/2008/06/tim_porter_on_formal_homotopy.html

Actually Porter gave two talks. The first was an introduction to simplicial methods and crossed complexes, but Bartlett didn't summarize that, and no slides are available. So for that, you should get ahold of the following free book:

30) Timothy Porter, The Crossed Menagerie: an Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology, available at http://www.informatics.bangor.ac.uk/~tporter/menagerie.pdf

and (harder) this review article highly recommended by Porter:

31) E. Curtis, Simplicial homotopy theory, Adv. Math. 6 (1971), 107-209.

The second talk by Porter, the one Bartlett blogged about, can be found at the workshop's website:

32) Timothy Porter, Formal homotopy quantum field theories and 2-groups, available at http://mat.uab.cat/~kock/crm/hocat/cat-groups/slides/Porter.pdf

This talk covered the papers by Martins, Porter and Turaev mentioned above.

I apologize to everyone whose talks I have not mentioned!

You can see more discussion of this Week's Finds at the n-Category Café.

Virtue is harmony. - attributed to Pythagoras

© 2008 John Baez