# August 11, 2007 {#week255}

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 ["Week 175"](#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.

::: {align="center"}
![](diary/paris_2007/paris_observatory_romer_speed_of_light.jpg)
:::

Later, Henri Poincaré 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 Poincaré's work
on relativity! Here's a good book that argues this case:

3) Peter Galison, Einstein's Clocks, Poincaré'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, `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`](http://arxiv.org/abs/math/0101038).

Daniel S. Freed, Twisted K-theory and loop groups, available as
[`arXiv:math/0206237`](http://arxiv.org/abs/math/0206237).

Daniel S. Freed, Michael J. Hopkins and Constantin Teleman, Loop groups
and twisted K-theory II, available as
[`arXiv:math/0511232`](http://arxiv.org/abs/math/0511232).

Daniel S. Freed, Michael J. Hopkins and Constantin Teleman, Twisted
K-theory and loop group representations, available as
[`arXiv:math/0312155`](http://arxiv/abs/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](https://arxiv.org/abs/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 \infty-category. The idea is to use diffeomorphisms and homotopies
between these as morphisms above dimension n. This gives an \infty-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
"(\infty,n)-category".

This sounds worse than an $n$-category, but it's okay for small n. In
particular, (\infty,1)-categories are pretty well understood by now. There
are a bunch of different approaches, with scary names like "topological
categories", "simplicial categories", "A~\infty~ categories", "Segal
categories", "complete Segal spaces", and "quasicategories".
Luckily, all these approaches are known to be equivalent - see
["Week 245"](#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 (\infty,1)-categories. In fact, starting as a grad
student, he wrote a mammoth tome generalizing topos theory from
categories to (\infty,1)-categories:

12) Jacob Lurie, Higher topos theory, available as
[`arXiv:math/0608040`](http://arxiv.org/abs/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
(\infty,1)-category version of 1Cob, and also the (\infty,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`](http://arxiv.org/abs/math/0605249).

And, Ulrike Tillmann gave a talk about this paper! It computes the
"nerve" of the (\infty,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 \infty-category into a
space, or simplicial set. (See item J of ["Week 117"](#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
(\infty,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 (\infty,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)\colon H(LM)^\otimesn^ \to H(LM)^\otimesm^

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 ["Week 220"](#week220).
Suitably punished, you can then enjoy this:

14) Ralph Cohen and Alexander Voronov, Notes on string topology,
available as [`arXiv:math/0503625`](http://arxiv.org/abs/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 as
[`arXiv:math/9911159`](http://arxiv.org/abs/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`](http://arxiv.org/abs/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 ["Week 197"](#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
["Week 149"](#week149) and ["Week 150"](#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)\colon Z(S) \to 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\colon S \to 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\colon M \to 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\colon S \to X;
-   a linear operator Z(M)\colon Z(S) \to Z(S') to any $n$-dimensional
    Riemannian cobordism M: S \to S' *equipped with a map* g\colon M \to 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\colon M \to T \times X

where we demand that the second component of this function is

g\colon M \to X

For example, if we've got a 1d QFT, we're letting a particle roam over
T \times 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) \to QFT(point)

This is called "pushing forward to a point".

More generally, we could hope that any map

F\colon X \to X'

gives a "pushforward" map

F~~*~~\colon QFT(X) \to 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\colon X \to C

Now suppose we map X to a point:

F\colon X \to point

What should the pushforward

F~~*~~\colon QFT(X) \to 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\colon X \to 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\colon X \to X'

with compact d-dimensional fibers, we can hope there's a pushforward
map

F~~*~~\colon QFT^p^(X) \to QFT^p-d^(X')

There should also be a pullback map

F^*^\colon QFT^p^(X') \to 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 σ 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\colon M \to X. The result -
usually called the "partition function" of the nonlinear σ 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 σ-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 σ-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\colon Sm(k)^op^ \to Simp\mathsf{Set}

where Simp\mathsf{Set} is the category of simplicial sets (see item C of
["Week 115"](#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 ∩ U_j.

Here "cover" means "cover in the Nisnevich topology" - that is, an
étale 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:

-   the projection maps X \times A^1 \to X
-   the maps C(U) \to 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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**Addendum:** For more discussion, go to the [$n$-Category Café](http://golem.ph.utexas.edu/category/2007/08/this_weeks_finds_in_mathematic_16.html).

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*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