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 a 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 L2 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
ΩB → 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 ΩB, 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,
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:
Starting with our extension
1 → F → E → B → 1
we pick a "section"
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
α: B → Aut(F)
where Aut(F) is the group of automorphisms of F. Here's how:
α(b) f = s(b) f s(b)-1
However, usually α 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: B2 → F
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 α. The sad thing about α is that it's not a homomorphism... but the good thing is, it's a homomorphism up to conjugation by g! In other words:
α(bb') f = g(b,b') [ α(b) α(b') f ] g(b,b')-1
Taken together, α 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:
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 α = 1. The set of isomorphism classes of central extensions is called H2(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 ExtΠ, H2Π and H2Π 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 for only spaces like this that the fundamental group is powerful enough to classify covering spaces. For more general spaces, we must use the fundamental groupoid. 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, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente, Lecture Notes in Mathematics 224, Springer, Berlin, 1971. Also available as math.AG/0206203.
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:
However, I'm not scared to go from n-groupoids to ω-groupoids, which are basically the same as spaces. In terms of spaces, the slogan goes like this:
15) Wikipedia, Grothendieck universe, http://en.wikipedia.org/wiki/Grothendieck_universe
Addendum: I'd like to thank Leo Alonso for pointing out that Grothendieck's famous "SGA1" is now available on the arXiv in TeX form, courtesy of the Société Mathématique de France. SGA2 is also available on the arXiv, and more are coming.
Here are further addenda thanks to Kevin Buzzard, Toby Bartels, David Corfield, Peter May, Jim Stasheff, and Ronnie Brown.
First, an email I sent in reply to Kevin Buzzard, who was curious about how we classify extensions of the group B by the group F using homomorphisms from B to the 2-group AUT(F). In particular, he wanted to know the definition of "2-group" and "homomorphism between 2-groups", and how AUT(F) is defined:
In particular, he wanted to know the definition of "2-group" and "homomorphism between 2-groups", and how AUT(F) is defined:
Dear Kevin -
You write:I'm just checking some of the details (extensions of groups are morphisms of 2-groups) and I find that you've not given me quite enough information to do it (in the sense that I'm not knowledgeable enough about standard facts about 2-groups to fill in some of the details that you allude to).Sorry. I'm glad you care enough to want to know this stuff: 2-groups and homomorphisms between them are defined in loving detail in my paper with Aaron Lauda, but I'll answer your questions here and append this to "week223" to help out anyone else who cares.Say we're in the "classical" case where F is an abelian group, B is a group, and we're classifying extensions 1→F→E→B→1 where F lies in the centre of E. We know the answer here: such E's are classified by H2(B,F) which, for me, means 2-cocycles over 2-coboundaries. Recall that a 2-cocycle is g:B2→F satisfying g(a,bc)+g(b,c)=g(a,b)+g(ab,c).Right.You want the answer to be homomorphisms B → AUT(F). You don't quite give the definitions of these things.True. Let me start by saying what a 2-group is, and then how the group B becomes a 2-group, and then how AUT(F) is defined.
A group is just a category with one object and with all morphisms invertible. Slick! But, we usually prefer to think of a group as a set: the morphisms of our category get called "elements" of this set. This set then has a multiplication function
m: G × G → G
and an identity element
1 ∈ G
satisfying the associative and unit laws, and such that every element has an inverse.
All this categorifies painlessly:
A 2-group is just a 2-category with one object and with all morphisms and 2-morphisms invertible. Slick! But, we sometimes prefer to think of a 2-group as a category: the morphisms of our 2-category get called "objects" of this category, and the 2-morphisms get called "morphisms". This category then has a multiplication functor
m: G × G → G
and an identity object
1 ∈ G
satisfying the associative and unit laws, and such that every object and morphism has an inverse.
In short: a 2-group is just like a group, but with the word "category" replacing the word "set", the word "object" replacing the word "element", and so on!
Now, how does a group become a 2-group? Simple: we take the elements of our group and make them the objects of our 2-group; then we say the only morphisms of our 2-group are identity morphisms. The 2-group multiplication m: G × G → G comes from the multiplication in our group, and so on.Let's stick to F abelian. You think of AUT(F) as being the 2-category with objects Aut(F), 1-morphisms Hom(φ,ψ)=F if φ=ψ and empty otherwise (because F is abelian). What are the 2-morphisms? Is Hom(φ,ψ) supposed to be a category with objects F?Alas, you're one dimension down: thought of as a 2-category, we want our 2-group AUT(F) to have one object, the usual group Aut(F) as morphisms, and conjugations between these as 2-morphisms.
Here's how we get this. Think of our group F as a category. Then let AUT(F) has F as its only object, invertible functors
a: F → F
as its morphisms, and natural isomorphisms between these as its 2-morphisms.
That's very slick. But let me say this in a different way using the other viewpoint, where we think of a 2-group as a "category with multiplication and inverses". Given a group F, our 2-group AUT(F) will be the category where the objects are automorphisms
a: F → F
and where a morphism f: a → a' is an element f of F that conjugates a to give a':
f a(g) f-1 = a'(g) for all g ∈ F
This viewpoint requires some extra work to check that AUT(F) is a 2-group. The 2-category viewpoint is actually much more efficient.Now what is a homomorphism B → AUT(F)?Here's where the subtlety comes in: weakening! So far we haven't weakened anything: all the equations in the definition of a group became equations in the definition of a 2-group. We're really just dealing with "strict" 2-groups here. But we need to weaken the definition of homomorphism, replacing some equations by isomorphisms, to get things to work out well now.
If we think of B and AUT(F) as 2-categories, a homomorphism B → AUT(F) is just a weak 2-functor. Slick! But, you may not enjoy this definition as much as I do.
So, let's think of B and AUT(F) as "categories with multiplication and inverses". Then a homomorphism
α: B → AUT(F)
is a functor that preserves multiplication of objects up to a specified isomorphism, which must satisfy some laws of its own.
Quite roughly, this means that given two objects b and b' of B, we don't have an equation
α(bb') = α(b) α(b')
Instead, we have an isomorphism
g(b,b'): α(bb') → α(b) α(b')
This needs to satisfy some equations. I can tell you these if you want, but for starters you can check that in our application, this g(b,b') thing will be the 2-cocycle familiar from group cohomology!
And, the laws g must satisfy say precisely that g is a 2-cocycle.
(Indeed, for a central extension α = 1, so all we really need to think about is this 2-cocycle g. Schreier theory goes on to consider more general extensions, where α ≠ 1.)I'm sure I could just go and read Breen's book, but these questions are so trivial that I'm sure you can instantly answer them, and you also get the confirmation that there are still people out there reading TWF.That's worth a lot! If you ever want to learn more about 2-groups and homomorphisms between them, I think my paper with Aaron Lauda could be easier than Breen's opus. Breen's opus concerns "higher Schreier" theory - classifying extensions of 2-groups with the help of 3-groups!
It's important to note that it's isomorphism classes of extensions that correspond to isomorphism classes of homomorphisms B → Aut(F). For this, one needs to know what a "2-isomorphism" between homomorphisms of 2-groups is. Again, this is explained in my paper with Lauda. It's a special case of a weak natural isomorphism between weak 2-functors between 2-categories, but we say what this means in terms that working mathematicians can understand.
Also, Toby Bartels had some comments on the dinosaurs of Cabazon and size issues in category theory:
John wrote in part:Thanks for clearing this up. I prefer not to distract people with size issues, so I didn't mention them until "the space of all spaces" walked in the door, at which point I figured alarm bells would start ringing for lots of people. But, it was already hiding in the "the (n+1)-groupoid of all n-groupoids". I prefer to use a new Grothendieck universe for each level of the n-categorical hierarchy, to justify such expressions. I guess ω-categories then require an ω-th universe.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.Did you see the creationist sign by the dinosaurs?
SPACES FIBERED OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE SPACE OF ALL SPACES 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.So if you want your slogan to treat size issues carefully:
SPACES FIBERED OVER THE SPACE B
ARE "THE SAME" AS
MAPS FROM B TO THE SPACE OF ALL SMALL SPACES
But you were secretly doing this all along! After all, when you wrote:n-GROUPOIDS FIBERED OVER AN n-GROUPOID B ARE "THE SAME" AS WEAK (n+1)-FUNCTORS FROM B TO THE (n+1)-GROUPOID nGpdyou simply used "nGpd" to abbreviate "OF ALL SMALL n-GROUPOIDS". So there are really no new size issues at the ω level.
(Exercise for those that like this sort of thing: Do we need to state that B and F are small?)
Yes, I saw that silly sign in front of the dinosaurs, though I didn't understand its full meaning until now. It wasn't there the last time I visited. Apparently the new owners have decided to enlist this nice roadside attraction as part of the propaganda campaign for creationism. They actually believe Adam and Eve walked with dinosaurs in Eden - as one biologist put it, "they think The Flintstones is a documentary".
It's sad how just as the magnificent history of the universe is becoming vividly clear, some want to truncate it to a pitifully human scale - and then claim that was God's work.
Next, David Corfield had some questions about the "space of all spaces", which I answered in this email:
Dear David -Next, Peter May wrote:
You wrote:Hi, SPACES FIBERED OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE SPACE OF ALL SMALL SPACES Is there another handle on this, other than OMEGA-GROUPOIDS FIBERED OVER THE OMEGA-GROUPOID B ARE "THE SAME" AS WEAK OMEGA-FUNCTORS FROM B TO THE OMEGA-GROUPOID OF ALL SMALL OMEGA-GROUPOIDS ? Presumably B must be small, and the spaces fibered over it.It suffices for the fibers to be small, so if you want a really nitpicky motto:
SMALL ω-GROUPOIDS FIBERED OVER THE ω-GROUPOID B
ARE "THE SAME" AS
WEAK ω-FUNCTORS FROM B TO THE ω-GROUPOID OF ALL
Do you and Jim have other intuitions about THE SPACE OF ALL SMALL SPACES?One can describe it in a completely precise and rigorous way. It's the disjoint union over all homotopy types of small spaces F of the classifying spaces B(Aut(F)). Here Aut(F) is the topological group of homotopy self-equivalences of F.
Note: the "largeness" of this space is solely due to it being a disjoint union of a proper class of connected components. When we map any small space to it, the map can only hit a set's worth of components. So, it's not really scary.
And, if we map a connected small space X to it, we get a map
X → B(Aut(F))
for some F, which is just what you need to get an F-bundle over X.Like, is it one of your FREE SUCH-AND-SUCHES?I don't know a description like that offhand, since "free" suggests a left universal property, and the space of all (small) spaces mainly has a right universal property, which describes maps into it.
Namely: maps from X into the space of all spaces are "the same as" fibrations over X.
I could make this completely precise, but it's probably not worth bothering here; one just needs suitable equivalence relations.
In his posting today, John Baez advertised the slogan:Here and below I've taken the liberty of numbering the references to papers, so it's easier to find them in my table of contents for This Week's Finds.
FIBRATIONS OVER THE BASE SPACE B WITH FIBER F
ARE "THE SAME" AS
HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F.
He hedged it with a "dose of vagueness", but in fact I proved a completely precise and general version of exactly this result in:
16) Peter May, Classifying spaces and fibrations, Memoirs AMS 155, Jan. 1975.
Using Moore loops on B, LB, one has a topological monoid, and one also has the topological monoid Aut(F) of homotopy equivalences of F. A "transport" is a homomorphism of topological monoids from LB to Aut(F). Allowing F to vary by a homotopy equivalence, one can define an equivalence relation on transports such that the equivalence classes are in natural bijective correspondence with the equivalence classes of "fibrations over the base space B with fiber F". One can generalize the context by allowing fibers in some nice category and prove the same result. See opus cit, Theorem 14.2, page 83. That was over 30 years ago, so naturally I wasn't thinking about categorification, but I would imagine that the methods categorify.
Jim Stasheff wrote:
Even more ancient:Ronnie Brown wrote:
17) James Stasheff, Parallel transport in fibre spaces, Bol. Soc. Mat. Mexicana (1968), 68-86.
Unfortunately that's a hard paper to get a hold of.
Somewhat related is:
18) James Stasheff, Associated fibre spaces, Michigan Math. Journal 15 (1968), 457-470.
and at the survey level:
19) James Stasheff, H-spaces and classifying spaces, I-IV, AMS Proc. Symp. Pure Math. 22 (1971), 247-272.
Of course, as you might expect, I describe things in terms of A∞-morphisms from the space of loops into Aut(F) of homotopy equivalences of F.
Now that some of us are comfortable with A∞-cats, categorification should proceed, perhaps with some technical details.
John Baez gave an interesting account of nonabelian cohomology and extension theory. Here are a few more references with which I have been involved, all using crossed complexes and free crossed resolutions:Finally, here's my reply to a bemused comment by Jim Stasheff:
20) Ronald Brown and P. J. Higgins, Crossed complexes and non-abelian extensions, Category theory proceedings, Gummersbach, 1981, (ed. K.H. Kamps et al) Lecture Notes in Math. 962 Springer, Berlin, 1982, pp. 39-50.
This generalises classical Schreier theory to extensions of groupoids.
21) Ronald Brown and O. Mucuk, Covering groups of non-connected topological groups revisited, Math. Proc. Camb. Phil. Soc., 115 (1994) 97-110. Also available as math.AT/0009021.
This relates the theory of covering topological groups of non connected topological groups to the classical theory of extensions and obstructions to a Q-kernel with an invariant in H3. It uses the properties of the internal hom in crossed complexes CRS(F,C), and exact sequences derived from a fibration C → D and the induced fibration on CRS(F, -).
22) Ronald Brown and Timothy Porter, On the Schreier theory of non-abelian extensions: generalisations and computations, Proceedings Royal Irish Academy 96A (1996), 213-227. Also available at http://www.informatics.bangor.ac.uk/public/math/research/ftp/algtop/rb/nonabex4.ps.gz
This establishes a generalisation of the Schreier theory in two ways (but only for groups). One is using coefficients in a crossed module, following Dedecker's key ideas, as in the references in John's account. Second it shows how to compute with such extensions
A → E → G
in terms of presentations of the group G. This involves identities among relations for the presentation, as shown originally by Turing in
23) Alan Turing, The extensions of a group, Compositio Mathematica 5 (1938), 357-367.
The advantage of this method is that one can actually do sums, even when the group G may be infinite. The example given by us is G = trefoil group with two generators x,y and relation x3=y2. This presentation has no identities among relations, and so the calculation is especially simple. Equivalence of extensions is described in terms of homotopies of morphisms of crossed complexes, and this relates the ideas to other forms of homological or homotopical algebra.
An advantage of this approach is the ability to calculate some small free crossed resolutions of some groups: this is one reason for using crossed complexes. Note that a convenient monoidal closed structure on the category of crossed complexes has been explicitly written down, and this allows convenient calculation and representations of homotopies, using the `unit interval' groupid, as a crossed complex.
One of the problems I have with the globular approach is the difficulty of writing down homotopies, and higher homotopies. For example, Ilhan Icen and I found it difficult to rewrite in terms of group-groupoids the well known Whitehead theory of automorphisms of crossed modules, explained for the crossed modules of groupoids case in:
24) Ronald Brown and Ilhan Icen, Homotopies and automorphisms of crossed modules of groupoids, Applied Categorical Structures, 11 (2003) 185-206. Also available as math.CT/0008117.
It looks as if it would be better expressed in terms of automorphisms of 2-groupoids: good marks to anyone who writes it down in that way!
One knows such homotopies of globular ∞ groupoids exist because globular ∞-groupoids are equivalent to crossed complexes:
25) Ronald Brown and P. J. Higgins, The equivalence of ∞-groupoids and crossed complexes, Cah. Top. Geom. Diff. 22 (1981) 371-386.
(This paper contains an early definition of a (strict, globular) ∞ category.)
This raises a question: what is the crossed complex associated to the free globular groupoid on one generator of dimension n? I have a round-about sketch proof, using also cubical theory, and a van Kampen theorem, that it is the fundamental crossed complex of the n-globe. Does anyone have a purely algebraic proof?
The idea of singular nonabelian cohomology of a space X with coefficients in a crossed complex C is given in:
26) Ronald Brown and P. J. Higgins, The classifying space of a crossed complex, Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120.
This cohomology is given by [ΠSX, C], homotopy classes of maps from the fundamental crossed complex of the singular complex of X, to C. There is also a Cech version (current work with Jim Glazebrook and Tim Porter).
An interesting problem is to classify extensions of crossed complexes!
There is an interesting account of extensions of principal bundles and transitive Lie groupoids by Androulidakis, developing work of Mackenzie, at:
27) Iakovos Androulidakis, Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel, J. Math. Phys. 45 (2004), 3095-4012. Also available as math.DG/0402007.
(not using crossed complexes).
Jim Stasheff wrote:John and anyone else who cares to weigh in, here are some comments from the purely topological or rather homotopy theory side: For both bundles and fibrations (e.g. over a paracompact base), your last slogan is the oldest: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) "the same as" referring to homotopy classes.It's certainly old, but I mentioned another that may be older:
COVERING SPACES OF B WITH FIBER F
ARE "THE SAME" AS
HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F
although one usually sees this special case (which I didn't bother to mention):CONNECTED COVERING SPACES OF B WITH FIBER F
ARE "THE SAME" AS
TRANSITIVE ACTIONS OF THE FUNDAMENTAL GROUP OF B ON F
which is usually disguised as follows:CONNECTED COVERING SPACES OF B
ARE "THE SAME" AS
SUBGROUPS OF THE FUNDAMENTAL GROUP OF B
Anyway, I wasn't trying to present things in historical order. I was trying present them roughly in order of increasing "dimension", starting with extensions of groups, then going up to 2-groups, then expanding out to groupoids, then going up to n-groupoids, and finally ω-groupoids... which are the same as homotopy types!
And here, as usual, the n-category theorists meet up with the topologists - and find that the topologists have already done everything there is to do with ω-groupoids ... but usually by thinking of them of them as spaces, rather than ω-groupoids!
It's sort of like climbing a mountain, surmounting steep cliffs with the help of ropes and other equipment, and then finding a Holiday Inn on top and realizing there was a 4-lane highway going up the other side.
So, as usual, the main point of calling homotopy types "ω-groupoids" instead of "spaces" is not to reinvent topology, but to see how ideas from topology generalize to n-category theory. Think of spaces as ω-groupoids, but use those as a springboard for ω-categories - or at least n-categories, perhaps just for low values of n if one is feeling tired.
In the case at hand, the ω-groupoidal slogan:FIBRATIONS OF ω-GROUPOIDS WITH FIBER F AND BASE B
ARE "THE SAME" AS
WEAK ω-FUNCTORS FROM B TO AUT(F)
is just a reformulation of:FIBRATIONS WITH FIBER F OVER THE SPACE B
ARE "THE SAME" AS
MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F)
but it suggests a grandiose generalization:
FIBRATIONS OF ω-CATEGORIES WITH BASE B
ARE "THE SAME" AS
WEAK ω-FUNCTORS FROM Bop TO THE ω-CATEGORY OF ω-CATEGORIES!
I guess we can thank Grothendieck for making precise and proving a version of this with ω replaced by n = 1:FIBRATIONS OF CATEGORIES WITH BASE B
ARE "THE SAME" AS
WEAK 2-FUNCTORS FROM Bop TO THE 2-CATEGORY OF CATEGORIES.
More recently people have been thinking about the n = 2 case, especially Claudio Hermida:
28) Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories, Applied Categorical Structures, 12 (2004), 427-459. Also available at http://maggie.cs.queensu.ca/chermida/papers/2-descent.pdf
He states something that hints at this:
FIBRATIONS OF 2-CATEGORIES WITH BASE B
ARE "THE SAME" AS
WEAK 3-FUNCTORS FROM Bop TO THE WEAK 3-CATEGORY OF 2-CATEGORIES.
Here I'm using Bop to mean B with the directions of both 1-morphisms and 2-morphisms reversed. Hermida follows tradition and calls this Bcoop - "op" for reversing 1-morphisms and "co" for reversing 2-morphisms. But, it looks like we'll be needing to reverse all kinds of morphisms in n-category case, so we'll need a short name for that.
© 2005 John Baez