Also available as http://math.ucr.edu/home/baez/week255.html
August 11, 2007
This Week's Finds in Mathematical Physics (Week 255)
John Baez
I've been roaming around Europe this summer - first Paris, then
Delphi and Olympia, then Greenwich, then Oslo, and now back to
Greenwich. I'm dying to tell you about the Abel Symposium in
Oslo. There were lots of cool talks about topological quantum
field theory, homotopy theory, and motivic cohomology.
I especially want to describe Jacob Lurie and Ulrike Tillman's
talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen's
talks on string topology, Stephan Stolz's talk on cohomology and
quantum field theory, and Fabien Morel's talk on A^1-homotopy
theory. But this stuff is sort of technical, and I usually try
to start each issue of This Week's Finds with something you don't
need a PhD to enjoy.
So, here's a tour of the Paris Observatory:
1) John Baez, Astronomical Paris,
http://golem.ph.utexas.edu/category/2007/07/astronomical_paris.html
Back when England and France were battling to rule the world,
each had a team of astronomers, physicists and mathematicians
devoted to precise measurement of latitudes, longitudes, and times.
The British team was centered at the Royal Observatory here in
Greenwich. The French team was centered at the Paris Observatory,
and it featured luminaries such as Cassini, Le Verrier and Laplace.
In "week175", written during an earlier visit to Greenwich, I
mentioned a book on this battle:
2) Dava Sobel, Longitude, Fourth Estate Ltd., London, 1996.
It's a lot of fun, and I recommend it highly.
There's a lot more to say, though. The speed of light was first
measured by Ole Romer at the Paris Observatory in 1676. Later,
Henri Poincare worked for the French Bureau of Longitude. Among
other things, he was the scientific secretary for its mission to
Ecuador.
To keep track of time precisely all over the world, you need to
think about the finite speed of light. This may have spurred
Poincare's work on relativity! Here's a good book that argues
this case:
3) Peter Galison, Einstein's Clocks, Poincare's Maps: Empires
of Time, W. W. Norton, New York, 2003. Reviewed by Robert Wald
in Physics Today at http://www.physicstoday.org/vol-57/iss-9/p57.html
I met Galison in Delphi, and it's clear he like to think about
the impact of practical stuff on math and physics.
I was in Delphi for a meeting of "Thales and Friends":
4) Thales and Friends, http://www.thalesandfriends.org
This is an organization that's trying to bridge the gap between
mathematics and the humanities. It's led by Apostolos Doxiadis,
who is famous for this novel:
5) Apostolos Doxiadis, Uncle Petros and Goldbach's Conjecture,
Bloomsbury, New York, 2000. Review by Keith Devlin at
http://www.maa.org/reviews/petros.html
There's a lot I could say about this meeting, but I just want
to advertise a forthcoming book by Doxiadis and a computer
scientist friend of his. It's a comic book - sorry, I mean
"graphic novel"! - about the history of mathematical logic
from Russell to Goedel:
6) Apostolos Doxiadis and Christos Papadimitriou, Logicomix,
to appear.
I saw a partially finished draft. I think it does a good job
of explaining to nonmathematicians what the big deal was with
mathematical logic around the turn of the last century... and
how these ideas eventually led to computers. It's also a fun
story.
If you're eager for summer reading and can't wait for Logicomix,
you might try this other novel by Papadimitrou:
7) Christos Papadimitriou, Turing (a Novel about Computation),
MIT Press, Boston, 2003.
It's a history of mathematics from the viewpoint of computer
science, as told by a computer program named Turing to a
lovelorn archaeologist. I haven't seen it yet.
Okay - enough fun stuff. On to the Abel Symposium!
8) Abel Symposium 2007, at http://abelsymposium.no/2007
Actually this was a lot of fun too. A bunch of bigshots were
there, including a bunch who didn't even give talks, like Eric
Friedlander, Ib Madsen, Jack Morava, and Graeme Segal.
(My apologies to all the bigshots I didn't list.)
Speaking of bigshots, Vladimir Voevodsky gave a special surprise
lecture on symmetric powers of motives. He wowed the audience not
only with his mathematical powers but also his ability to solve a
technical problem that had stumped all the previous speakers! The
blackboards in the lecture hall were controlled electronically,
by a switch. But, the blackboards only moved a few inches before
stalling out. So, people had to keep hitting the switch over and
over. It was really annoying, and it became the subject of running
jokes. People would ask the speakers: "Can't you talk and press
buttons at the same time?"
So, what did Voevodsky do? He lifted the blackboard by hand!
He laughed and said "Russian solution". But, I think it's a great
example of how he gets around problems by creative new approaches.
It really pleased me how many talks mentioned n-categories, and
even used them to do exciting things. This seems quite new. In
the old days, bigshots might think about n-categories, but they'd
be embarrassed to actually mention them, since they had a
reputation for being "too abstract".
In fact, Dan Freed alluded to this in his talk on topological
quantum field theory. He said that every mathematician has
an "n-category number". Your n-category number is the largest n
such that you can think about n-categories for a half hour without
getting a splitting headache.
When Freed first invented this concept, he felt pretty
self-satisfied, since his n-category number was 1, while for
most mathematicians it was 0. But lately, he says, other
people's n-category numbers have been increasing, while his has
stayed the same.
He said this makes him suspicious. In light of the scandals
plaguing the Tour de France and American baseball, he suspects
mathematicians are taking "category-enhancing substances"!
Freed shouldn't feel bad: he was among the first to introduce
n-categories in the subject of topological quantum field theory!
He gave a nice talk on this, clear and unpretentious, leading
up to a conjecture for the 3-vector space that Chern-Simons
theory assigns to a point.
That would make a great followup to these papers on the 2-vector
space that Chern-Simons theory assigns to a circle:
9) Daniel S. Freed, The Verlinde algebra is twisted equivariant
K-theory, available as arXiv:math/0101038.
Daniel S. Freed, Twisted K-theory and loop groups, available
as arXiv:math/0206237.
Daniel S. Freed, Michael J. Hopkins and Constantin Teleman,
Loop groups and twisted K-theory II, available as
arXiv:math/0511232.
Daniel S. Freed, Michael J. Hopkins and Constantin Teleman,
Twisted K-theory and loop group representations, available as
arXiv:math/0312155.
In a similar vein, Jacob Lurie talked about his work with Mike
Hopkins in which they proved a version of the "Baez-Dolan cobordism
hypothesis" in dimensions 1 and 2. I'm calling it this because
that's what Lurie called it in his title, and it makes me feel good.
You can read about this hypothesis here:
10) John Baez and James Dolan, Higher-dimensional algebra and
topological quantum field theory, J.Math.Phys. 36 (1995) 6073-6105
Also available as arXiv:q-alg/9503002.
It was an attempt to completely describe the algebraic structure of
the n-category nCob, where:
objects are 0d manifolds,
1-morphisms are 1d manifolds with boundary,
2-morphisms are 2d manifolds with corners,
3-morphisms are 3d manifolds with corners,
.
.
.
and so on up to dimension n. Unfortunately, at the time we
proposed it, little was known about n-categories above n = 3.
For a more recent take on these ideas, see:
11) Eugenia Cheng and Nick Gurski, Towards an n-category of
cobordisms, Theory and Applications of Categories 18 (2007)
274-302. Available at
http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html
Lurie and Hopkins use a new trick: they redefine nCob to be a
special sort of infinity-category. The idea is to use
diffeomorphisms and homotopies between these as morphisms above
dimension n. This gives an infinity-category version of nCob,
where:
objects are 0-dimensional manifolds,
1-morphisms are 1-dimensional manifolds with boundary,
2-morphisms are 2-dimensional manifolds with corners,
3-morphisms are 3-dimensional manifolds with corners,
.
.
.
n-morphisms are n-dimensional manifolds with corners,
(n+1)-morphisms are diffeomorphisms,
(n+2)-morphisms are homotopies between diffeomorphisms,
(n+3)-morphisms are homotopies between homotopies,
.
.
.
and so on for ever!
Since everything here is invertible above dimension n, this is
called an "(infinity,n)-category".
This sounds worse than an n-category, but it's okay for small
n. In particular, (infinity,1)-categories are pretty well
understood by now. There are a bunch of different approaches,
with scary names like "topological categories", "simplicial
categories", "A_infinity categories", "Segal categories",
"complete Segal spaces", and "quasicategories". Luckily,
all these approaches are known to be equivalent - see
"week245" for some good introductory material by Julie
Bergner and Andre Joyal. Joyal is now writing a book on
this stuff.
Lurie is a real expert on (infinity,1)-categories. In fact,
starting as a grad student, he wrote a mammoth tome generalizing
topos theory from categories to (infinity,1)-categories:
12) Jacob Lurie, Higher topos theory, available as
arXiv:math/0608040.
I'm sure Freed would suspect him of taking category-enhancing
substances: his category number is infinite, and this book is
619 pages long! Then he went on to apply this stuff to algebraic
geometry... and the world is still reeling. I was happy to
discover that he's a nice guy, enthusiastic and friendly - not
the terrifying fiend I expected.
Anyway, Lurie and Hopkins have worked out the precise structure
of the (infinity,1)-category version of 1Cob, and also the
(infinity,2)-category version of 2Cob. Unfortunately this work
is not yet written up. But, they use results from this paper:
13) Soren Galatius, Ib Madsen, Ulrike Tillmann, Michael Weiss,
The homotopy type of the cobordism category, available as
arXiv:math/0605249
And, Ulrike Tillman gave a talk about this paper! It computes
the "nerve" of the (infinity,1)-category where:
objects are (n-1)-dimensional manifolds,
1-morphisms are n-dimensional manifolds with boundary,
2-morphisms are diffeomorphisms,
3-morphisms are homotopies between diffeomorphisms,
4-morphisms are homotopies between homotopies,
.
.
.
The "nerve" is a trick for turning any sort of infinity-category
into a space, or simplicial set. (See item J of "week117" for
the nerve of a plain old category. This should give you the
general idea.)
In her talk, she went further and computed the nerve of
the (infinity,k)-category where:
objects are (n-k)-dimensional manifolds,
1-morphisms are (n-k+1)-dimensional manifolds with boundary,
2-morphisms are (n-k+2)-dimensional manifolds with corners,
.
.
.
k-morphisms are n-dimensional manifolds with corners,
(k+1)-morphisms are diffeomorphisms,
(k+2)-morphisms are homotopies between diffeomorphisms,
(k+3)-morphisms are homotopies between homotopies,
.
.
.
This is also joint work with the same coauthors, but it
seems not to be written up yet, except for k = 1, where it's
proved in the above paper. The cool thing about the new work
is that it uses an idea familiar from higher category theory - a
k-simplicial space - to give a rigorous description of the
nerve of the above (infinity,k)-category! Indeed, Tillmann
told me she thinks of k-simplicial spaces as just a convenient
way of dealing with higher categories.
Stephan Stolz's talk also involved cobordism n-categories,
but I'll say more about that later.
Ralph Cohen and Dennis Sullivan both gave talks on string
topology - a trick for studying a space by studying collections
of loops in that space, and relating this to ideas from string
theory.
String topology started when Chas and Sullivan took the ideas
of string theory and applied them in a somewhat ethereal form to
strings propagating in any manifold.
In full-fledged string theory, one of the main tools is "conformal
field theory". In a CFT, if you have a state of n strings, and a
Riemann surface going from n strings to m strings, you get a state
of m strings.
A good way to get CFTs is to consider strings propagating on some
manifold or other. Of course the manifold needs some sort of
geometry, like a Riemannian metric, for your strings to know how
to propagate.
But Chas and Sullivan figured out what you can do if the spacetime
is a bare manifold, without any metric. Basically, you just
need to stick the word "homology" in front of everything! This
makes everything sufficiently floppy.
So, instead of considering actual loops in a manifold M,
which form a space LM, they took the homology of LM and got
a vector space or abelian group H(LM). Then, for each homology
class C on the moduli space of Riemann surfaces that go from n
circles to m circles, they got an operation with n inputs and m
outputs:
Z(C): H(LM)^{tensor n} -> H(LM)^{tensor m}
All these operations fit together into a slight generalization of
an operad, called a "PROP".
If you don't remember what an "operad" is, give yourself twenty
lashes with a wet noodle and review "week220". Suitably punished,
you can then enjoy this:
14) Ralph Cohen and Alexander Voronov, Notes on string topology,
available as arXiv:math/0503625.
Both PROPs and operads are defined near the beginning here.
PROPs and operads are gadgets for describing operations with
any number of inputs. Operads can only handle operations with
one output. PROPS can handle operations with any number of outputs.
To see a more geometrical treatment of string topology, the way
it looked before the operadchiks got ahold of it, try the original
paper by Chas and Sullivan:
15) Moira Chas and Dennis Sullivan, String topology,
available asarXiv:math/9911159.
Sullivan talked about some recent refinements of string topology
which deal with the fact that the moduli space of Riemann surfaces
has a "boundary", so it doesn't have a closed "top-dimensional
homology class".
Cohen's talk described some cool relations between string topology
and symplectic geometry! In physics we use symplectic manifolds to
describe the space of states - the so-called "phase space" - of a
classical system. So, if you have a loop in a symplectic
manifold, it can describe a periodic orbit of some classical
system. In particular, if we pick a periodic time-dependent
Hamiltonian for this system, a loop will be a solution of
Hamilton's equations iff it's a critical point for the "action".
But, we can also imagine letting loops move in the direction of
decreasing action, following the "gradient flow". They'll trace
out 2d surfaces which we can think of as string world-sheets!
This is just what string topology studies, but now we can get
"Morse theory" into the game: this studies a space (here LM)
by looking at critical points of a function on this space, and
its gradient flow.
So, we get a nice interaction between periodic orbits in phase
space, and the string topology of that space, and Morse theory!
For more, try this:
16) Ralph Cohen, The Floer homotopy type of the cotangent bundle,
available as arXiv:math/0702852.
Next, let me say a bit about Stephan Stolz's talk. He spoke
on his work with Peter Teichner, which is a very ambitious
attempt to bring quantum field theory right into the heart of
algebraic topology.
I discussed this in "week197". I said they were working
on a wonderful analogy between quantum field theories and
different flavors of cohomology. It's been published since
then:
17) Stephan Stolz and Peter Teichner, What is an elliptic object?
Available at http://math.berkeley.edu/~teichner/papers.html
Back then, the analogy looked like this:
1-dimensional supersymmetric QFTs complex K-theory
2-dimensional supersymmetric conformal QFTs elliptic cohomology
When I saw this, I tried to guess a generalization to higher
dimensions.
There's an obvious guess for the right-hand column, since there's
something called the "chromatic filtration", which is - very
roughly - a list of cohomology theories. Complex K-theory
is the 1st entry on this list, and elliptic cohomology is the 2nd!
(For a lot more details, see "week149" and "week150".)
There's also an obvious guess for the left-hand column:
n-dimensional supersymmetric QFTs of some sort!
The problem is the word "conformal" in the second row. In
2 dimensions, a conformal structure is a way of making
spacetime look locally like the complex plane. This is great,
because elliptic cohomology has a lot to do with complex
analysis - or more precisely, elliptic curves and modular forms.
But, it's not clear how one should generalize this to higher
dimensions!
Luckily, thanks to a subsequent conversation with Witten,
Stolz and Teichner realized that the partition function of a
2d supersymmetric QFT gives a modular form even if the QFT is
not invariant under conformal transformations. This means we
can remove the word "conformal" from the second row! For more
details, try this:
18) Stephan Stolz and Peter Teichner, Super symmetric field
theories and integral modular forms, preliminary version
available at http://math.berkeley.edu/~teichner/papers.html
They've also gone back and added a 0th row to their chart.
It's always wise to start counting at zero! Now the chart
looks much nicer:
0-dimensional supersymmetric QFTs deRham cohomology
1-dimensional supersymmetric QFTs complex K-theory
2-dimensional supersymmetric QFTs elliptic cohomology
Yes, good old deRham cohomology is the 0th entry in the
"chromatic filtration"! It's the least scary sort of cohomology
theory, at least for physicists. They get scarier as we move
down the chart.
Quantum field theory also gets scarier as we move down the chart -
the infinities that plague quantum field theory tend to get worse
in higher dimensions of spacetime. So, while we can dream about
extensions of this chart, there's already plenty to handle here.
The most audacious idea in Stolz and Teichner's work is to take
a manifold X and study the set of all n-dimensional QFT's
"parametrized by X".
For X a point, such a thing is just an ordinary n-dimensional
QFT. Roughly speaking, this is a gadget Z that assigns:
a Hilbert space Z(S) to any (n-1)-dimensional Riemannian
manifold S;
a linear operator Z(M): Z(S) -> Z(S') to any n-dimensional
Riemannian manifold M going from S to S'.
If you're a mathematician, you may know that M is really a
"cobordism" from S to S', written M: S -> S'. And if you're
really cool, you'll know that cobordisms form a symmetric
monoidal category nCob, and that Z should be a symmetric
monoidal functor.
If you're a physicist, you'll know that S stands for "space"
and "M" stands for "spacetime". All the stuff I'm describing
should remind you of the definition of a "TQFT", except now our
spaces and spacetimes have Riemannian metrics, because we're
doing honest QFTs, not topological ones.
Given a spacetime M, we try to compute the operator Z(M) as
a path integral; for example, an integral over all maps
f: M -> T
where f is a "field" taking values in a "target space" T.
If this seems too scary, take n = 1. Then we've got a
1-dimensional quantum field theory, so we can take our
spacetime M to be an interval. Then f is just a path in some
space T. In this case the path integral is really an integral
over all paths a particle could trace out in T. So, 1-dimensional
quantum field theory is just ordinary quantum mechanics!
There are a lot of subtleties I'm skipping over here, both on
the math and physics sides. But never mind - the really cool
part is this generalization:
Roughly speaking, an n-dimensional QFT "parametrized by X"
assigns:
a Hilbert space Z(S) to any (n-1)-dimensional Riemannian
manifold S EQUIPPED WITH A MAP g: S -> X;
a linear operator Z(M): Z(S) -> Z(S') to any n-dimensional
Riemannian cobordism M: S -> S' EQUIPPED WITH A MAP g: M -> X.
If you're a mathematician, you may see we've switched to using
cobordisms "over X". It's a straightforward generalization.
But what does it mean physically? Here the path integral picture
is helpful. Now we're doing a path integral over all fields
f: M -> T x X
where we demand that the second component of this function is
g: M -> X
For example, if we've got a 1d QFT, we're letting a particle
roam over T x X, but demanding that its X coordinates follow
a specific path g.
So, we're doing a *constrained* path integral!
In heaven, everything physicists do can be made mathematically
rigorous. Up there, knowing how to do these constrained path
integrals would tell us how to do unconstrained path integrals:
we'd just integrate over all choices of the path g. So, a QFT
parametrized by X would automatically give us an ordinary QFT.
Now, an ordinary QFT is just a QFT parametrized by a point!
So, if we use QFT(X) to mean the set of n-dimensional QFTs
parametrized by X, we'd have a map
QFT(X) -> QFT(point)
This is called "pushing forward to a point".
More generally, we could hope that any map
F: X -> X'
gives a "pushforward" map
F_*: QFT(X) -> QFT(X')
Let's see if this makes any sense. In fact, I've been
overlooking some important issues. An example will shed light
on this.
Consider a 0-dimensional QFT parametrized by some manifold X.
Let's call it Z. What is Z like, concretely?
For starters, notice that the only (-1)-dimensional manifold
is the empty set. A 0-dimensional manifold "going from the
empty set to the empty set" is just a set of points. Also,
while I didn't mention it earlier, all manifolds in this game
must be *compact*. So, this set of points must be finite.
If you now take the definition I wrote down and use that
"symmetric monoidal functor" baloney, you'll see Z assigns a
*number* to any finite set of points mapped into X. Furthermore,
this assignment must be multiplicative. So, it's enough to know
a number for each point in X. In short, our QFT is just a
function:
Z: X -> C
Now suppose we map X to a point:
F: X -> point
What should the pushforward
F_*: QFT(X) -> QFT(point)
do to the function Z?
There's an obvious guess: we should *integrate* this function
on X to get a number - that is, a function on a point. Indeed,
that's what "path integration" should reduce to in this
pathetically simple case: plain old integration!
Alas, there's no good way to integrate a function over X unless
this manifold comes equipped with a measure. But, if X is
compact, oriented and p-dimensional, we can integrate a *p-form*
over X.
More generally, if we have a bundle
F: X -> X'
with compact d-dimensional fibers, we can take a p-form on X
and integrate it over the fibers to get a (p-d)-form on X'.
This is how you "push forward" differential forms.
So, pushing forward is a bit subtler than I led you to believe at
first. We should really talk about n-dimensional QFTs "of degree
p" parametrized by X. Let's call the set of these
QFT^p(X)
I won't define them, but for n = 0 they're just p-forms on X.
Anyway: if we have a bundle
F: X -> X'
with compact d-dimensional fibers, we can hope there's a
pushforward map
F_*: QFT^p(X) -> QFT^{p-d}(X')
There should also be a pullback map
F^*: QFT^p(X') -> QFT^p(X)
This is a lot less tricky, and I'll let you figure out how
it works.
I should warn you, I've been glossing over lots of important
aspects of this work - like the role played by n-categories,
and the role played by supersymmetry. Supersymmetry doesn't
matter much for the broad conceptual picture I've been sketching.
But, we need it for this analogy to work:
0-dimensional supersymmetric QFTs deRham cohomology
1-dimensional supersymmetric QFTs complex K-theory
2-dimensional supersymmetric QFTs elliptic cohomology
The idea is to impose an equivalence relation on supersymmetric
QFTs, called "concordance", and try to show:
The set of concordance classes of degree-p 0d supersymmetric
QFTs parametrized by X is the pth de Rham cohomology group of X.
The set of concordance classes of degree-p 1d supersymmetric
QFTs parametrized by X is the pth K-theory group of X.
The set of concordance classes of degree-p 2d supersymmetric
QFTs parametrized by X is the pth elliptic cohomology group of X.
So far people have done this in the 0d and 1d cases. The 2d
case is a major project, because it pushes the limits of what
people can do with quantum field theory.
Why did I spend so much time talking about pushforwards of
QFTs? Well, it's very important for defining invariants like
the "fundamental class" of an oriented manifold, or the "A-hat
genus" of a spin manifold, or the "Witten index" of a string
manifold.
Here's how it goes, very roughly. Suppose X is a compact
Riemannian manifold. Then the simplest n-dimensional QFT
parametrized by X is the one where we take the target space T
(mentioned a while back) to be just a point!
This parametrized QFT is called the "nonlinear sigma model",
for stupid historic reasons. All the fun happens when we push
this QFT forwards to a point. Then we integrate over all the
maps g: M -> X. The result - usually called the "partition
function" of the nonlinear sigma model - should be an interesting
invariant of X.
In the case n = 1, this trick gives the "A-hat genus" of X,
but it only works when X is a spin manifold: we need this to
define the 1d supersymmetric nonlinear sigma-model.
In the case n = 2, this trick gives the "Witten genus" of X,
but it only works when X is a string manifold: we need this to
define the 2d supersymmetric nonlinear sigma-model.
For more on the n = 1 case, see:
19) Henning Hohnhold, Peter Teichner and Stephan Stolz, From
minimal geodesics to super symmetric field theories. In
memory of Raoul Bott. Available at
http://math.berkeley.edu/~teichner/papers.html
For the n = 2 case, see the papers I already listed.
(I'm confused about the case n = 0, for reasons having to do
with the "degree" I mentioned earlier.)
Finally: the cool part, which I haven't even mentioned, is
that we really need to describe n-dimensional QFTs using an
*n-category* of cobordisms - not just a mere 1-category, as
I sloppily said above.
This first gets exciting when we hit n = 2: you'll see a bunch
of stuff about 2-categories (or technically, "bicategories")
in the old Stolz-Teichner paper "What is an elliptic object",
listed above.
In short: we're starting to see a unified picture where we
study spaces by letting particles, strings, and their
n-dimensional cousins roam around in these spaces.
There are lots of slight variants: string topology, the
Stolz-Teichner picture, and of course good old-fashioned
topological quantum field theory. All of them have a lot
to do with n-categories.
There's a lot more to say about all this... but luckily, there
should be a proceedings of this conference, where you can read
more. My own talk is here:
20) John Baez, Higher gauge theory and elliptic cohomology,
http://math.ucr.edu/home/baez/abel/
It, too, is about studying spaces by letting strings roam
around inside them!
But instead of summarizing my own talk, I want to say a
bit about the other side of the symposium - the motivic
cohomology side!
I'll only summarize a few basic definitions. I got these from
the talks by Fabian Morel and Vladimir Voevodsky, and I want
to write them down before I forget! For more, try these:
21) Fabian Morel and Vladimir Voevodsky, A^1-homotopy theory
of schemes, September 1998. Available at
http://citeseer.ist.psu.edu/morel98suphomotopy.html
22) Vladimir Voevodsky (notes by Charles Weibel),
Voevodsky's Seattle Lectures: K-theory and motivic
cohomology. Available at http://citeseer.ist.psu.edu/249068.html
Okay:
A^1-homotopy theory is an attempt to do homotopy theory for
algebraic geometry. In algebraic geometry we often work over
a fixed field k, and the goal here is to create a category
which contains smooth algebraic varieties over k as objects,
but also other more general spaces, providing a sufficiently
flexible category in which to do homotopy theory.
One of the simplest smooth algebraic varieties over k is the
"affine line" A^1. The algebraic functions on this line are
just polynomials in one variable with coefficients in k. In
A^1-homotopy theory, we want to set up a context where we can
use the affine line A^1 to parametrize homotopies, much as we
use the unit interval [0,1] in ordinary homotopy theory.
For this, people start by looking at Sm(k), the category of
smooth algebraic varieties over k. Then, they consider
the category of "simplicial presheaves" on Sm(k).
A simplicial presheaf on Sm(k) is just a functor
F: Sm(k)^{op} -> SimpSet
where SimpSet is the category of simplicial sets (see item C of
"week115") We think of F as specifying some sort of space by
telling us for each smooth algebraic variety X the simplicial set
F(X) of all maps into this space.
To make this kind of abstract space work nicely, F(X) should
depend "locally" on X. For this, we insist that given a cover
of a variety X by varieties U_i, guys in F(X) are the same as
guys in F(U_i) that agree on the intersections
U_i intersect U_j.
Here "cover" means "cover in the Nisnevich topology" - that is,
an etale cover such that every point being covered is the image
of a point in the cover for which the covering map induces an
isomorphism of residue fields.
If you've come this far, you may not be scared to hear that the
Nisnevich topology is really a "Grothendieck topology" on Sm(k),
and I'm really demanding that F be a "sheaf" with respect to this
topology.
So, the kind of "space" we're studying is a simplicial sheaf on
the category of smooth varieties over k with its Nisnevich
topology. We call these category of these guys Space(k).
Just saying this already makes me feel smart. Just think how
smart I'd feel if I knew why the Nisnevich topology was better
than the good old etale topology!
Anyway, to do homotopy theory with these simplicial sheaves, we
need to make Space(k) into a "model category". I should have
explained model categories in some previous Week, but I've never
gotten around to it, and right now is not the time. So, I'll
just say one key thing.
The *most* important thing about a model category is that
it's equipped with a collection of morphisms that act
like homotopy equivalences. They're called "weak equivalences".
Already in ordinary topology, these weak equivalences are a
slight generalization of homotopy equivalences. They're
actually the same as homotopy equivalences when the spaces
involved are nice; they're designed to work better for nasty
spaces.
In A^1 homotopy theory, the weak equivalences are generated
by two kinds of morphisms:
A) the projection maps X x A^1 -> X
B) the maps C(U) -> X coming from covers U of X.
Here X is any space in Space(k), and C(U) is the "Cech nerve"
of the cover U.
This framework seems like a really cool blend of algebraic
geometry and homotopy theory. But, to do homology theory
in a good way we need to go a bit further, and introduce "motives".
However, I'm tired, and I bet you are too! Motives are a big
idea, and it doesn't make sense to start talking about them now.
So, some other day....
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Quote of the Week:
And if a bird can speak, who once was a dinosaur, and a dog can dream,
should it be implausible that a man might supervise the construction
of light? - King Crimson
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Addendum: For more discussion, go to the
http://golem.ph.utexas.edu/category/2007/08/this_weeks_finds_in_mathematic_16.html
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If you just want the latest issue, go to
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