From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2874 Path: news.gmane.org!not-for-mail From: "John Baez" Newsgroups: gmane.science.mathematics.categories Subject: Schreier theory Date: Mon, 14 Nov 2005 13:19:36 -0800 (PST) Message-ID: <200511142119.jAELJaD28664@math-cl-n03.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018957 6185 80.91.229.2 (29 Apr 2009 15:29:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:17 +0000 (UTC) To: categories@mta.ca (categories) Original-X-From: rrosebru@mta.ca Wed Nov 16 17:11:01 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 16 Nov 2005 17:11:01 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EcUPt-0005A3-EG for categories-list@mta.ca; Wed, 16 Nov 2005 17:02:05 -0400 X-Mailer: ELM [version 2.5 PL6] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 19 Original-Lines: 586 Xref: news.gmane.org gmane.science.mathematics.categories:2874 Archived-At: Dear Categorists - This issue of my column has an introduction to nonabelian cohomology that some of you might enjoy. You'll have to skip past some other stuff to get to it. Best, jb ........................................................................ Also available as http://math.ucr.edu/home/baez/week223.html November 14, 2005 This Week's Finds in Mathematical Physics - Week 223 John Baez This week I'd like to talk about two aspects of higher gauge theory: p-form electromagnetism and nonabelian cohomology. Lurking behind both of these is the mathematics of n-categories, but I'll do my best to hide that until the end, to build up the suspense. But first, some cool pictures. Astronomy is booming these days, and it's a great way to see beautiful complexity emerging from simple laws in this wonderful universe of ours. So, I'd like the freedom to occasionally start This Week's Finds with some pictures from the skies. Think of it as an appetizer before the main course. Sometimes I'll explicitly relate these pictures to math and physics; other times not. Here's Saturn's moon Hyperion, photographed up close by the Cassini probe: 1) Cassini-Huyghens Mission, Hyperion: Odd World, http://saturn.jpl.nasa.gov/multimedia/images/image-details.cfm?imageID=1762 It seems to be a huge pile of rubble loosely held together by gravity and heavily cratered by meteor bombardments. Hyperion is interesting because it's the only known moon that tumbles chaotically on a short time scale, thanks to its eccentric shape and gravitational interactions with Saturn and Titan. This leads to some interesting math. We can think of Hyperion's angular momentum vector as a point on a sphere. If we started out knowing this point lay inside some small disk, time evolution would warp this disk into an ever more complicated region as time passed. This region would always have the same area, thanks to the wonders of symplectic geometry. But it would sprout ever more complicated tendrils, with its perimeter growing by a factor of e about every 100 days or so! That's chaos for you. Indeed, only quantum mechanics would stop the intricacy from growing forever, by blurring it out. After about 37 years, the area of a typical tendril would equal Planck's constant. At this point, classical mechanics would no longer be accurate. You'd really need to describe Hyperion's spin state using quantum theory: for example, a holomorphic section of some line bundle on the sphere. Well... at least you would if it weren't for decoherence caused by the interaction of Hyperion with its environment, for example solar radiation! For an explanation of how this changes the story, try: 2) Michael Berry, Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?), in Quantum Mechanics: Scientific Perspectives on Divine Action, CTNS Publications, Vatican Observatory, 2001. Also available at http://www.phy.bris.ac.uk/people/berry_mv/the_papers/berry337.pdf Here's another great picture: 3) The Hubble Heritage Project, Cat's Eye Nebula - NGC 6543, http://heritage.stsci.edu/2004/27/index.html This is a star about the size of the Sun, nearing the end of its life, emitting pulses of gas and dust. Astronomers call such a thing a "planetary nebula", though it has nothing to do with planets. It's in our galaxy, about 3000 light years from us. When it's done shedding its outer layers, all that's left of this star will be a white dwarf. Our own Sun will become a planetary nebula in about 6.9 billion years, after two separate stages of being a red giant - one as it runs out hydrogen, and one as it runs out of helium. When the helium is all gone, the Sun will start to pulsate every 100,000 years, ejecting more and more mass in each pulse, eventually throwing off all but the hot inner core made of heavier elements. The astronomer Bruce Balick has written eloquently on what this will mean for the Earth: Here on Earth, we'll feel the wind of the ejected gases sweeping past, slowly at first (a mere 5 miles per second!), and then picking up speed as the spasms continue - eventually to reach 1000 miles per second!! The remnant Sun will rise as a dot of intense light, no larger than Venus, more brilliant than 100 present Suns, and an intensely hot blue-white color hotter than any welder's torch. Light from the fiendish blue "pinprick" will braise the Earth and tear apart its surface molecules and atoms. A new but very thin "atmosphere" of free electrons will form as the Earth's surface turns to dust. So, don't keep procrastinating - enjoy life now! For other pictures of planetary nebulae, try Balick's webpage: 4) Bruce Balick, Hubble Space Telescope images of planetary nebulae, http://www.astro.washington.edu/balick/WFPC2/index.html For a timeline of the universe, including the future life of our Sun, try: 5) John Baez, A brief history of the universe, http://math.ucr.edu/home/baez/timeline.html Now... on to p-form electromagnetism! In ordinary electromagnetism, the secret star of the show turns out to be not the electromagnetic field but the "vector potential", A. At least locally, we can think of this as a 1-form. A 1-form is just a gadget that you can integrate along a path and get a number. In the case of the vector potential, this number describes the change in phase that a charged particle acquires as it moves along this path. The 1-form A gives rise to a 2-form F called the "electromagnetic field". A 2-form is a gadget you can integrate over a surface and get a number. Here's how we get F from A. Suppose we move a charged particle around a loop that's the boundary of some surface. Then the integral of F over this surface is defined to be the integral of A around the loop! We summarize this by saying that F is the "exterior derivative" of A, and writing F = dA. F is called the electromagnetic field because... that's what it is! It contains both the electric and magnetic fields in a single neat package. In 4d spacetime, the magnetic field describes the change in a phase of a charged particle that loops around a surface in the xy, yz or zx planes. The electric field describes the change in phase of a charged particle that loops around a surface in the xt, yt or zt planes. If you don't know this stuff, you're missing some of the best fun life has to offer. For an easy introduction with lots of gorgeous pictures, see: 6) Derek Wise, Electricity, magnetism and hypercubes, available at http://math.ucr.edu/~derek/talks/050916bw.pdf The idea of p-form electromagnetism is to replace point particles by strings or higher-dimensional membranes. To see how this goes, it's enough to look at 2-form electromagnetism. In 2-form electromagnetism, the star of the show is a 2-form, A. As already mentioned, a 2-form is a gadget you can integrate over a surface and get a number. In 2-form electromagnetism, this number describes the change in phase that a charged string acquires as it moves along, tracing out a surface in spacetime. The 2-form A gives rise to a 3-form, F. A 3-form is a gadget you can integrate over a 3-dimensional region and get a number. Suppose we move a charged string and let it trace out a surface that's the boundary of some 3-dimensional region. Then the integral of F over this region is defined to be the integral of A over the surface! Again we write this as: F = dA. So, we're just adding one to the dimensions of things. This makes it easy to keep on going. In fact, for any integer p, we can write down a generalization of Maxwell's equations. It goes like this. We start with a p-form A. We define a (p+1)-form F = dA This automatically implies some of Maxwell's equations: dF = 0 but the nontrivial Maxwell equations say that *d*F = J where * is the Hodge star operator and J is a p-form called the "current", which is produced by charged matter. What does this mean, physically? The idea is that we have charged matter consisting of (p-1)-dimensional membranes. These trace out p-dimensional surfaces in spacetime as time passes. The current J is a p-form that's concentrated on these surfaces. The current affects the A field in a manner governed by Maxwell's equations. Conversely, the A field affects the motion of the membranes. Classically, we just integrate the A field over the surface traced out by a membrane and add the result to the *action* for the membrane. In the path integral approach to quantum mechanics, this number gives a change in phase, as already mentioned. Maxwell's equations and their p-form generalization make sense when spacetime is any Lorentzian manifold. However, to get a theory where initial data determine a unique global solution, we want our spacetime to be "globally hyperbolic", which means that it has a "Cauchy surface": roughly, a spacelike surface that any sufficiently long timelike curve hits precisely once. To get a good *quantum* theory of p-form electromagnetism with a Hilbert space of states on which time evolution acts as unitary operators, we need more: our spacetime should be "stationary", meaning that it has time translation symmetry. Otherwise there's no way to define energy and the vacuum state - which is defined to be the least-energy state. My student Miguel Carrion-Alvarez tackled an important special case in his thesis, namely "static" globally hyperbolic spacetimes: 7) Miguel Carrion-Alvarez, Loop quantization versus Fock quantization of p-form electromagnetism on static spacetimes, available as math-ph/0412032. There's a lot of interesting analysis involved, especially when space (the Cauchy surface) is noncompact. When it's compact, we can use "Hodge's theorem" to relate its deRham cohomology to its topology, and this turns out to be crucial for understanding p-form electromagnetism - especially issues like the p-form Bohm-Aharonov effect. When it's noncompact we need something called "twisted L^2 cohomology" instead, and Miguel proved a generalization of Hodge's theorem for this. With the analysis under control, Miguel was able to set up a very beautiful approach to "loop quantum electromagnetism" and its p-form generalization. Here the idea is to write Maxwell's equations in terms of the integrals of A around all possible loops in space - or more generally, over all p-dimensional surfaces. People interested in loop quantum gravity should like this. As you can guess, either from seeing all the "d" operators or seeing all the buzzwords I'm throwing around, p-form electromagnetism is really just cohomology incarnated as physics! My student Derek Wise made this very precise for a version of the theory where spacetime is *discrete* - so-called "lattice p-form electromagnetism": 8) Derek Wise, Lattice p-form electromagnetism and chain field theory, available as gr-qc/0510033. Version with better graphics and related material at http://math.ucr.edu/~derek/pform/index.html In this paper, he shows lattice p-form electromagnetism is a "chain field theory": something like a topological quantum field theory, but where what matters is not spacetime itself so much as the cochain complex of differential forms *on* spacetime, equipped with just enough extra geometrical structure to write down the p-form version of Maxwell's equations. Both Miguel's thesis and Derek's papers are great if you want to learn lots of math and physics. I seem to attract students who enjoy explaining things. Speaking of which.... Next I want to explain some stuff Danny Stevenson told me at a mall in the little town of Cabazon while we were recovering from a hike in the desert followed by pancakes at the Wheel Inn - a roadside restaurant famous for its enormous statues of dinosaurs. Danny works on gerbes, stacks, and higher gauge theory. Last year we wrote a paper with Alissa Crans and Urs Schreiber constructing 2-groups (categorified groups) from the math of string theory - more precisely, from central extensions of loop groups. Since then I've been spending a lot of time writing a paper with Urs on higher gauge theory, where we set up a theory of parallel transport along surfaces. 2-form electromagnetism is the simplest case of this theory. Meanwhile, Danny has been thinking about connections on 2-vector bundles and their relation to the cohomology of Lie 2-algebras. This has led him to generalize Schreier theory in some interesting ways. So, let me tell you about Schreier theory! Schreier theory is a way to classify short exact sequences of groups. I'll say what I mean by that in a minute... but what makes Schreier theory special is that avoids some simplifying assumptions you might have seen if you've studied short exact sequences before. Normally people water down their short exact sequences by assuming some of the groups in question are *abelian*. This lets them use "cohomology theory" to do the classification. See "week210" for a nice book that takes this approach. This standard approach is great - I'm not knocking it - but Schreier theory is more general: it's really a branch of "nonabelian cohomology theory". It's not all that hard to explain, either. So, I'll explain it and then talk about various simplifying assumptions people make. The goal of Schreier theory is to classify short exact sequences of groups: 1 -> F -> E -> B -> 1 for a given choice of F and B. "Exact" means that the arrows stand for homomorphisms and the image of each arrow is the kernel of the next. Here this just means that F is a normal subgroup of E and B is the quotient group E/F. Such a short exact sequence is also called an "extension of B by F", since E is bigger than B and contains F. The simplest choice is to let E be the direct sum of F and B. Usually there are other more interesting extensions as well. To classify these, the trick is to use the analogy between group theory and topology. As I explained in "week213", you can think of a group as a watered-down version of a connected space with a chosen point. The reason is that given such a space, we get a group consisting of homotopy classes of loops based at the chosen point. This is called the "fundamental group" of our space. There's a lot more information in our space than this group. But pretty much anything you can do for groups, you can do for such spaces. It's usually harder, but it's completely analogous! In particular, classifying short exact sequences is a lot like classifying "fibrations": 1 -> F -> E -> B -> 1 where now the letters stand for connected spaces with a chosen point, and the arrows stand for continuous maps. If you're a physicist or geometer you may prefer fiber bundles to "fibrations" - but luckily, they're so similar we can ignore the difference in a vague discussion like this. The idea is basically just that E maps onto B, and sitting over each point of B we have a copy of F. We call B the "base space", E the "total space" and F the "fiber". If we want to classify such fibrations we can consider carrying the fiber F around a loop in B and see how it twists around. For example, if all our spaces are smooth manifolds, we can pick a connection on the total space E and see what parallel transport around a loop in the base space B does to points in the fiber F. This gives a kind of homomorphism Omega(B) -> Aut(F) sending loops in B to invertible maps from F to itself. And, the cool thing is: this homomorphism lets us classify the fibration! Here I say "kind of homomorphism" since Omega(B), the space of loops in B based at the chosen point, is only "kind of" a topological group: the group laws only hold up to homotopy. But let's not worry about this technicality - especially since I'm being vague about all sorts of other equally important issues! The reason I can get away with not worrying about these issues is that I'm trying to explain a very robust powerful principle - one that can easily survive a dose of vagueness that would kill a lesser idea. Namely, if B is a connected space with a chosen basepoint, FIBRATIONS OVER THE BASE SPACE B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F. This could be called "the basic principle of Galois theory", for reasons explained in "week213". There I explained the special case where the fiber is discrete. Then our fibration called a "covering space", and the basic principle of Galois theory boils down to this: COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F. Okay. Now let's use the same principle to classify extensions of a group B by a group F: 1 -> F -> E -> B -> 1 The group B here acts like "loops in the base". But what acts like "automorphisms of the fiber"? You might guess it's the group of automorphisms of F. But, it's actually the *2-group* of automorphisms of F! A 2-group is a categorified version of a group where all the usual group laws hold up to natural isomorphism. They play a role in higher gauge theory like that of groups in ordinary gauge theory. In higher gauge theory, parallel transport along a path is described by an *object* in a 2-group, while parallel transport along a path-of-paths is described by a *morphism*. In 2-form electromagnetism we use a very simple "abelian" 2-group, which has one object and either the real line or the circle as morphism. But there are other more interesting "nonabelian" examples. If you want to learn more about 2-form electromagnetism from this perspective, try "week210". For 2-groups in general, try this paper: 9) John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-groups, Theory and Applications of Categories 12 (2004), 423-491. Available online at http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html or as math.QA/0307200. Anyway: it turns out that any group F gives a 2-group AUT(F) where the objects are automorphisms of F and the morphisms are "conjugations" - elements of F acting to conjugate one automorphism and yield another. And, extensions 1 -> F -> E -> B -> 1 are classified by homomorphisms B -> AUT(F) where we think of B as a 2-group with only identity morphisms. More precisely: EXTENSIONS OF THE GROUP B BY THE GROUP F ARE "THE SAME" AS HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F) It's fun to work out the details, but it's probably not a good use of our time together grinding through them here. So, I'll just sketch how it works. Starting with our extension i p 1 --> F --> E --> B --> 1 we pick a "section" s E <-- B meaning a function with p(s(b)) = b for all b in B. We can find a section because p is onto. However, the section usually *isn't* a homomorphism. Given the section s, we get a function alpha: B -> Aut(F) where Aut(F) is the group of automorphisms of F. Here's how: alpha(b) f = s(b) f s(b)^{-1} However, usually alpha *isn't* a homomorphism. So far this seems a bit sad: functions between groups want to be homomorphisms. But, we can measure how much s fails to be a homomorphism using the function g: B^2 -> F given by g(b,b') = s(bb') s(b')^{-1} s(b)^{-1} Note that g = 1 iff s is a homomorphism. We can then use this function g to save alpha. The sad thing about alpha is that it's not a homomorphism... but the good thing is, it's a homomorphism up to conjugation by g! In other words: alpha(bb') f = g(b,b') [alpha(b) alpha(b') f] g(b,b')^{-1} Taken together, alpha and g satisfy some equations ("cocycle conditions") which say precisely that they form a homomorphism from B to the 2-group AUT(F). Conversely, any such homomorphism gives an extension of B by F. In fact, isomorphism classes of extensions of B by F correspond in a 1-1 way with isomorphism classes of homorphisms from B to AUT(F). So, we've classified these extensions! In fact, something even better is true! It's evil to "decategorify" by taking isomorphism classes as we did in the previous paragraph. To avoid this, we can form a groupoid whose objects are extensions of B by F, and a groupoid whose objects are homomorphisms B -> AUT(F). I'm pretty sure that if you form these groupoids in the obvious way, they're equivalent. And that's what this slogan really means: EXTENSIONS OF THE GROUP B BY THE GROUP F ARE "THE SAME" AS HOMOMORPHISMS FROM B TO THE 2-GROUP AUT(F) Next, let me say how Schreier theory reduces to more familiar ideas in two special cases. People have thought a lot about the special case where F is abelian and lies in the center of E. These are called "central extensions". This is just the case where alpha = 1. The set of isomorphism classes of central extensions is called H^2(B,F) - the "second cohomology" of B with coefficients in F. People have also thought about "abelian extensions". That's an even more special case where all three groups are abelian. The set of isomorphism classes of such extensions is called Ext(B,F). Since we don't make any simplifying assumptions like this in Schreier theory, it's part of a subject called "nonabelian cohomology". It was actually worked out by Dedecker in the 1960's, based on much earlier work by Schreier: 10) O. Schreier, Ueber die Erweiterung von Gruppen I, Monatschefte fur Mathematik and Physick 34 (1926), 165-180. Ueber die Erweiterung von Gruppen II, Abh. Math. Sem. Hamburg 4 (1926), 321-346. 11) P. Dedecker, Les foncteuers Ext_Pi, H^2_Pi and H^2_Pi non abeliens, C. R. Acad. Sci. Paris 258 (1964), 4891-4895. More recently, Schreier theory was pushed one step up the categorical ladder by Larry Breen. As far as I can tell, he essentially classified the extensions of a 2-group B by a 2-group F in terms of homomorphisms B -> AUT(F), where AUT(F) is the *3-group* of automorphisms of F: 12) Lawrence Breen, Theorie de Schreier superieure, Ann. Sci. Ecole Norm. Sup. 25 (1992), 465-514. Also available at http://www.numdam.org/numdam-bin/feuilleter?id=ASENS_1992_4_25_5 We can keep pushing Schreier theory upwards like this, but we can also expand it "sideways" by replacing groups with groupoids. You should have been annoyed by how I kept assuming my topological spaces were connected and equipped with a specified point. I did this to make them analogous to groups. For example, it's only spaces like this for which the fundamental group is sufficiently powerful to classify covering spaces. For more general spaces, we should use the fundamental *groupoid* instead of the fundamental group. And, we can set up a Schreier theory for extensions of groupoids: 13) V. Blanco, M. Bullejos and E. Faro, Categorical non abelian cohomology, and the Schreier theory of groupoids, available as math.CT/0410202. In fact, these authors note that Grothendieck did something similar back in 1971: he classified *all* groupoids fibered over a groupoid B in terms of weak 2-functors from B to Gpd, which is the 2-groupoid of groupoids! The point here is that Gpd contains AUT(F) for any fixed groupoid F: 14) Alexander Grothendieck, Categories fibrees et descente (SGA I), Lecture Notes in Mathematics 224, Springer, Berlin, 1971. Having extended the idea "sideways" like this, one can then continue marching "upwards". I don't know how much work has been done on this, but the slogan should be something like this: n-GROUPOIDS FIBERED OVER AN n-GROUPOID B ARE "THE SAME" AS WEAK (n+1)-FUNCTORS FROM B TO THE (n+1)-GROUPOID nGpd Grothendieck also studied this kind of thing with categories replacing groupoids, so there should also be an n-category version, I think... but it's more delicate to define "fibrations" for categories than for groupoids, so I'm a bit scared to state a slogan suitable for n-categories. However, I'm not scared to go from n-groupoids to omega-groupoids, which are basically the same as spaces. In terms of spaces, the slogan goes like this: SPACES FIBERED OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE SPACE OF ALL SPACES This is how James Dolan taught it to me. Most mortals are scared of "the space of all spaces" - both for fear of Russell's paradox, and because we really need a *space* of all spaces, not just a mere set of them. To avoid these terrors, you can water down Jim's slogan by choosing a specific space F to be the fiber: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) where AUT(F) is the topological group of homotopy self-equivalences of F. The fearsome "space of all spaces" is then the disjoint union of the classifying spaces of all these topological groups AUT(F). It's too large to be a space unless you pass to a larger universe of sets, but otherwise it's perfectly fine. Grothendieck invented the notion of a "Grothendieck universe" for precisely this purpose: 14) Wikipedia, Grothendieck universe, http://en.wikipedia.org/wiki/Grothendieck_universe ----------------------------------------------------------------------- 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/twf.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