
I'm at this workshop now, and I want to talk about it:
1) Workshop on Categorical Groups, June 1620, 2008, Universitat de Barcelona, organized by Pilar Carrasco, Josep Elgueta, Joachim Kock and Antonio Rodríguez Garzón, http://mat.uab.cat/~kock/crm/hocat/catgroups/
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 twentyfive 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 "neoPythagoreans". 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 neoPythagoreans, 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 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:
FIRE hot dry AIR EARTH wet cold WATER
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 dodecahedronThis 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?
A modernday 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 curledup 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 nicesounding 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 socalled circle of fifths doesn't quite close, since
(3/2)^{12} / 2^{7} = 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, "equaltempered") 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 builtin conflict is the spice of life! It makes it impossible for harmony to be perfect and therefore dull.
Anyway, Pythagorean tuning is not equaltempered: 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 nicesounding 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:
1 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:
1 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 "2group", 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 2groups. 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 2groups, 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 2bundles" with a given "gauge 2group". It's a famous old theorem that for any topological group G, we can find a space BG such that principal Gbundles 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 2groups!
Indeed, Baas Bökstedt and Kro did something much more general for topological 2categories:
10) Nils Baas, Marcel Bökstedt and Tore Kro, 2Categorical Ktheories, available as arXiv:math/0612549.
Just as a group is a category with one object and with all morphisms being invertible, a 2group is a 2category with one object and all morphisms and 2morphisms invertible. But the 2group 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 2group, available as arXiv/0801.3843
and that's what I talked about. If you're also interested in classifying spaces of 2categories that aren't topological, just "discrete", you should try these:
12) John Duskin, Simplicial matrices and the nerves of weak ncategories I: nerves of bicategories, available at http://www.tac.mta.ca/tac/volumes/9/n10/910abs.html
13) Manuel Bullejos and A. Cegarra, On the geometry of 2categories 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 2categories, Applied Categorical Structures 13 (2005), 223233. Also available as arXiv:math/0406615.
On Monday afternoon, Bruce Bartlett spoke on a geometric way to understand representations and "2representations" 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 2representations of finite groups and their 2characters, 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 2representations 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 2representations of finite groups and their 2characters, 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:
S / \ / \ 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 2Hilbert 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 DijkgraafWitten model.
Next, to handle the TQFT called ChernSimons 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 2group. Then he'd make contact with the categorified DijkgraafWitten TQFT studied in these papers:
17) David Yetter, TQFT's from homotopy 2types, Journal of Knot Theory and its Ramifications 2 (1993), 113123.
18) Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, I: Formal maps and crossed Calgebras, 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, 375403. Also available as arXiv:math/0512034.
19) João Faria Martins and Timothy Porter, On Yetter's invariant and an extension of the DijkgraafWitten 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 2group G.
Given all this, it's natural to contemplate a further generalization of Bruce's work where G is a Lie 2group. Unfortunately, Lie 2groups don't have many representations on 2Hilbert space of the sort I've secretly been talking about so far: that is, finitedimensional ones.
So we may, perhaps, need to ponder representations of Lie 2groups on infinitedimensional 2Hilbert 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 2groups 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 infinitedimensional 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 infinitedimensional representations, along with the intertwining operators between them... and the "2intertwining 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 2group of a topological space.
To set these in their proper perspective, it's good to recall the periodic table of ncategories, mentioned in "week49":
ktuply monoidal ncategories n = 0 n = 1 n = 2 k = 0 sets categories 2categories k = 1 monoids monoidal monoidal categories 2categories k = 2 commutative braided braided monoids monoidal monoidal categories 2categories k = 3 " " symmetric sylleptic monoidal monoidal categories 2categories k = 4 " " " " symmetric monoidal 2categories k = 5 " " " " " "
The idea here is that an (n+k)category with only one jmorphism for j < k acts like an ncategory with extra bells and whistles: a "ktuply monoidal ncategory". 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 ncategory with everything invertible is called an "ngroupoid". Such a thing is believed to be essentially the same as a "homotopy ntype", 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 ngroupoid version of the Periodic Table can be understood using homotopy theory. It looks like this:
ktuply groupal ngroupoids n = 0 n = 1 n = 2 k = 0 sets groupoids 2groupoids k = 1 groups 2groups 3groups k = 2 abelian braided braided groups 2groups 3groups k = 3 " " symmetric sylleptic 2groups 3groups k = 4 " " " " symmetric 3groups k = 5 " " " " " "
Most of this workshop has focused on 2groups. 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 2groups, 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 onetoone, while its cokernel says if it's onto. Functions can be nice in two basic ways: they can be onetoone, 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, MartínezMoreno and Enrico Vitale, Chain complexes of symmetric categorical groups, JPAA 196 (2005), 279312. Also available at http://www.math.ucl.ac.be/membres/vitale/SCGcompl3.pdf
21) Pilar Carrasco, Antonio Garzón and Enrico Vitale, On categorical crossed modules, TAC 16 (2006), 85618, available as http://tac.mta.ca/tac/volumes/16/22/1622abs.html
By generalizing properties of the category of abelian groups, people invented the concept of "abelian category". Similarly, Vitale told us a definition of "2abelian 2category", obtained by generalizing properties of the 2category of symmetric 2groups. 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 ngroups 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 ngroupoids, Ph.D. Thesis, Università di Milano, 2008. Also available at arXiv:0802.0800.
The relation between 2groups and topology is made explicit using the concept of "fundamental 2group". Just as every space equipped with a basepoint has a fundamental group, it has a fundamental 2group. And for a homotopy 2type, this 2group 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 2group of a space for this workshop. Unfortunately, he was unable to come. Luckily, you can still see his talk:
24) David Roberts, Fundamental 2groups and 2covering 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 "2covering spaces" of a connected space are classified by "sub2groups" of its fundamental 2group!
Aurora del Río spoke on fundamental 2groups and their application to Ktheory. Whenever we have a fibration of pointed spaces
F → E → B
we get a long exact sequence of homotopy groups
... → π_{n}(F) → π_{n}(E) → π_{n}(B) → π_{n1}(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 π_{n}(X), is just the fundamental group of the (n1)fold loop space of X. So, the Spanish categorical group experts define the nth "homotopy 2group" of a space X to be the fundamental 2group of an iterated loop space of X. And, it turns out that any fibration of spaces gives a long exact sequence of homotopy 2groups!
I was surprised by this, but in retrospect I shouldn't have been. Any fibration gives a "long exact sequence of iterated loop spaces":
... → L^{n}F → L^{n} E → L^{n} B → L^{n1}F → ...
So, as soon as we have a definition of "fundamental ngroupoids" and long exact sequences of ngroupoids, and can show that taking the fundamental ngroupoid preserves exactness, we can get a long exact sequence of fundamental ngroupoids. If we simply define a fundamental ngroupoid to be a homotopy ntype, this should not be hard.
But this was just the warmup for Aurora's talk, which was about Ktheory. Quillen set up modern algebraic Ktheory by defining the Kgroups of a ring R to be the homotopy groups of a certain space called BGL(R)^{+}. In here talk, Aurora defined the K2groups of a ring in the same way, but using homotopy 2groups! 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 Ktheory for categorical groups, http://mat.uab.cat/~kock/crm/hocat/catgroups/slides/delRio.pdf
See also the paper she and Antonio Garzón wrote on this topic:
26) Antonio Garzón and Aurora del Río, On algebraic Ktheory categorical groups, http://www.ugr.es/~agarzon/KthCG.pdf
Also try these other papers:
26) Antonio Garzón and Aurora del Río, Lowdimensional cohomology of categorical groups, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 (2003), 247280. Available at http://www.numdam.org/numdambin/fitem?id=CTGDC_2003__44_4_247_0
This one gets into Ktheory:
27) Antonio Garzón and Aurora del Río, The Whitehead categorical group of derivations, Georgian Mathematical Journal 09 (2002), 709721. 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 2groups (in the guise of crossed modules). Luckily Tim Porter summarized it at the nCategory Café:
27) Timothy Porter, Behrang Noohi on butterflies and weak morphisms between 2groups, 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 2groups, available as arXiv:math/0506313.
Also at the nCategory 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 2groups, 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), 107209.
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 2groups, available at http://mat.uab.cat/~kock/crm/hocat/catgroups/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 nCategory Café.
Virtue is harmony.  attributed to Pythagoras
© 2008 John Baez
baez@math.removethis.ucr.andthis.edu
