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, Revetements Etales et Groupe Fondamental
(SGA1), chapter VI: Categories fibrees 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:
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:
15) Wikipedia, Grothendieck universe,
http://en.wikipedia.org/wiki/Grothendieck_universe
-----------------------------------------------------------------------
Quote of the week:
"Hah, what a fantastic night," Gunn said. "Arcturus is absolutely steady."
He leaned back, put his elbows on the rail of the lift, and looked up at
the sky. His glasses glinted faintly in the starlight. "Astronomy is not
terribly important," he said. He fell silent for a moment. "Although it
is one of the more important things we do as a species," he said. He did
not see any contradiction there. - Richard Preston, First Light
-----------------------------------------------------------------------
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
Societe Mathematique 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:
Dear Kevin -
> 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 knowledgable
> 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, mentioned above, 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 H^2(B,F)
> which, for me, means 2-cocycles over 2-coboundaries. Recall that
> a 2-cocycle is g:B^2-->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 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 x G -> G
and an identity element
1 in 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 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 x G -> G
and an identity object
1 in 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 x 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(phi,psi)=F if phi=psi and empty otherwise
> (because F is abelian). What are the 2-morphisms? Is Hom(phi,psi) 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 in 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
alpha: 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
alpha(bb') = alpha(b) alpha(b')
Instead, we have an isomorphism
g(b,b') : alpha(bb') -> alpha(b) alpha(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 alpha = 1, so all we really need
to think about is this 2-cocycle g. Schreier theory goes on to consider
more general extensions, where alpha is not 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!
Best,
jb
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.
Next, Toby Bartels had some comments on the dinosaurs of Cabazon and
size issues in category theory:
John wrote in part:
> 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 nGpd
you simply used "nGpd" to abbreviate "OF ALL SMALL n-GROUPOIDS".
So there are really no new size issues at the omega level.
(Exercise for those that like this sort of thing:
Do we need to state that B and F are small?)
-- Toby
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 handle such expressions. I guess omega-categories
then require an omega-th universe.
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 -
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 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
> 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 sentence completely precise, but it's probably not
worth bothering here; one just needs suitable equivalence relations.
Next, Peter May wrote:
In his posting today, John Baez advertised the slogan:
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.
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.
Jim Stasheff wrote:
Even more ancient:
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_\infty-morphisms from the space of loops into Aut(F) of homotopy
equivalences of F.
Now that some of us are comfortable with A_\infty-cats,
categorification should proceed, perhaps with some technical details.
jim
Ronnie Brown wrote:
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:
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 as
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 x^3=y^2. 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 infinity groupoids exist because
globular infinity-groupoids are equivalent to crossed complexes:
25) Ronald Brown and P. J. Higgins, The equivalence of $\infty$-groupoids
and crossed complexes, Cah. Top. Geom. Diff. 22 (1981), 371-386.
(This paper contains an early definition of a (strict, globular)
infinity 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 [\Pi SX, 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).
Ronnie Brown
www.bangor.ac.uk/r.brown
Finally, here's my reply to a bemused comment by Jim Stasheff:
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 OVER 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 OVER 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 OVER 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 omega-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 omega-groupoids ... but usually by thinking of
them of them as *spaces*, rather than omega-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 "omega-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
omega-groupoids but use those as a springboard for omega-categories -
or at least n-categories, perhaps just for low values of n if one is
feeling tired.
In the case at hand, the omega-groupoidal slogan:
FIBRATIONS OF OMEGA-GROUPOIDS WITH FIBER F AND BASE B
ARE "THE SAME" AS
WEAK OMEGA-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 OMEGA-CATEGORIES WITH BASE B
ARE "THE SAME" AS
WEAK OMEGA-FUNCTORS FROM B^{op} TO THE OMEGA-CATEGORY OF OMEGA-CATEGORIES!
I guess we can thank Grothendieck for making precise and proving a
version of this with omega replaced by n = 1:
FIBRATIONS OF CATEGORIES WITH BASE B
ARE "THE SAME" AS
WEAK 2-FUNCTORS FROM B^{op} 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 B^{op} TO THE WEAK 3-CATEGORY OF 2-CATEGORIES.
where I'm using B^{op} to mean B with the directions of both 1-morphisms
and 2-morphisms reversed.
(Hermida follows tradition and calls this B^{coop} - "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.)
Best,
jb
-----------------------------------------------------------------------
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