Also available as http://math.ucr.edu/home/baez/week256.html
August 27, 2007
This Week's Finds in Mathematical Physics (Week 256)
John Baez
My European wanderings continue. I'm in Greenwich again, just back
from a mindblowing conference in Vienna, part of a bigger program
that's still going on:
1) Poisson sigma models, Lie algebroids, deformations, and higher
analogues, Erwin Schroedinger Institute, August  September 2007,
organized by Thomas Strobl, Henrique Bursztyn, and Harald Grosse.
Program at http://w3.impa.br/~henrique/esi.html
I learned a huge amount, both from the talks and from conversations
with Urs Schreiber and others. Mainly, I learned that I've really
been falling behind the times when it comes to classical mechanics
and quantization!
I could easily spend several Weeks trying to assimilate the
halfdigested information I acquired and explain it all to you.
But, I want to get back to the Tale of Groupoidification! So,
I'll only say a little about this wonderful conference.
You may know that in classical mechanics, the space of states of
a physical system is called its "phase space". Often this is described
by a "symplectic manifold"  a manifold equipped with a nondegenerate
closed 2form. Sometimes it's described by a "Poisson manifold" 
a manifold equipped with a bracket operation on its smooth
functions, making the smooth functions into a Lie algebra and
also satisfying the product rule:
{f,gh} = {f,g}h + g{f,h}
Every symplectic manifold gives a Poisson manifold, but not vice
versa. A good example of a Poisson manifold that's not symplectic
is the phase space of a spinning point particle, which has angular
momentum but no other properties.
Every mathematical physicist should know some symplectic geometry
and Poisson geometry! To get started on symplectic geometry, try
these, in rough order of increasing difficulty:
1) Vladimir I. Arnold, Mathematical Methods of Classical Mechanics,
Springer, Berlin, 1997.
2) Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics,
BenjaminCummings, New York, 1978.
3) Victor Guillemin and Shlomo Sternberg, Symplectic Techniques
in Physics, Cambridge U. Press, Cambridge, 1990.
4) Ana Cannas da Silva, Symplectic geometry, available as
arXiv:math.SG/0505366.
5) Sergei Tabachnikov, Introduction to symplectic topology,
available at http://www.math.psu.edu/tabachni/courses/symplectic.pdf
For Poisson geometry, try the above but also:
6) Alan Weinstein, Poisson geometry, available at
http://galileo.stmarysca.edu/bdavis/poisson.pdf
7) Darryl Holm, Applications of Poisson geometry to physical
problems, available as arXiv:0708.1585.
8) I. Vaisman, Lectures on the Geometry of Poisson Manifolds,
Birkhaeuser, Boston, 1994.
All this stuff is great. But lately, people have started thinking
about generalizations of the idea of phase space that go far beyond
Poisson manifolds! In fact there seems to be an infinite sequence,
which begins like this:
symplectic manifolds,
Poisson manifolds,
Courant algebroids,
...
I'd heard of Courant algebroids before, but they always seemed
like a scary and arbitrary concept  until I came across this
paper in Vienna:
9) Pavol Severa, Some title containing the words "homotopy"
and "symplectic", e.g. this one, available as arXiv:math/0105080.
The title is goofy, but the paper itself contains some truly
visionary speculations. Among other things, it argues that the
above sequence of concepts really goes like this:
symplectic manifolds,
symplectic Lie algebroids,
symplectic Lie 2algebroids,
symplectic Lie 3algebroids,
...
These, in turn, are infinitesimal versions of perhaps more fundamental
concepts:
symplectic manifolds,
symplectic Lie groupoids,
symplectic Lie 2groupoids,
symplectic Lie 3groupoids,
...
These concepts take the basic concept of classical phase space and
build in symmetries, symmetries between symmetries, and so on!
So, we may be starting to see the "periodic table of ncategories"
show up in classical mechanics. Back in "week49" I explained the
most basic version of this table. Here's a tiny portion of it:
ktuply monoidal ncategories
n = 0 n = 1 n = 2
k = 0 sets categories 2categories
k = 1 monoids monoidal monoidal
categories 2categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2categories
k = 3 " " symmetric sylleptic
monoidal monoidal
categories 2categories
k = 4 " " " " symmetric
monoidal
2categories
k = 5 " " " " " "
An ncategory has objects, 1morphisms betwen objects, 2morphisms
between 1morphisms, and so on up to the nth level. A "ktuply
monoidal" ncategory is an (n+k)category that's trivial on the
bottom k levels. It masquerades as an ncategory with extra bells
and whistles. As you can see, we get lots of fun structures this
way.
The concept of ncategory is very general: it describes things,
processes that go between things, metaprocesses that go between
processes and so on. But, in classical mechanics we may want
to demand that all these morphisms be invertible, and that all the
ways of composing them be smooth functions. Then we should get some
table like this:
ktuply groupal Lie ngroupoids
n = 0 n = 1 n = 2
k = 0 manifolds Lie groupoids Lie 2groupoids
k = 1 Lie groups Lie 2groups Lie 3groups
k = 2 abelian braided braided
Lie groups Lie 2groups Lie 3groups
k = 3 " " symmetric sylleptic
Lie 2groups Lie 3groups
k = 4 " " " " symmetric
Lie 3groups
k = 5 " " " " " "
There are lots of technical issues to consider  for example, whether
manifolds are a sufficiently general notion of "smooth space" to
make this chart really work. But for now, the key thing is to
understand what we're shooting for, so we can set up definitions that
accomplish it.
For example, it would be nice if we could "differentiate" any of the
gadgets on the above table, just as we differentiate a Lie group
and get a Lie algebra. This should give another table, like this:
ktuply groupal Lie nalgebroids
n = 0 n = 1 n = 2
k = 0 vector bundles? Lie algebroids Lie 2algebroids
k = 1 Lie algebras Lie 2algebras Lie 3algebras
k = 2 abelian braided braided
Lie algebras Lie 2algebras Lie 3algebras
k = 3 " " symmetric sylleptic
Lie 2algebras Lie 3algebras
k = 4 " " " " symmetric
Lie 3algebras
k = 5 " " " " " "
The n = k = 0 corner is a bit puzzling  it's sort of degenerate.
Everyone knows how to get Lie algebras from Lie groups. So, the
real fun starts in getting Lie algebroids from Lie groupoids!
If you want to see how it works, start here:
10) Alan Weinstein, Groupoids: unifying internal and external
symmetry, AMS Notices, 43 (1996), 744752. Also available as
arXiv:math/9602220
For more details, try this:
11) Kirill Mackenzie, General Theory of Lie Groupoids and Lie
Algebroids, Cambridge U. Press, 2005.
There's also the question of going back. We can integrate any
finitedimensional Lie algebra to get a simplyconnected Lie group 
that's called Lie's 3rd theorem. But getting from Lie algebroids
to Lie groupoids is harder... in fact, according to the standard
definitions, it's often impossible!
That's bad enough, but the really weird part is this: you can
get something like a Lie *2groupoid* from a Lie algebroid!
This throws a serious monkey wrench into the whole periodic
table.
Luckily, one of the people who really understands this stuff
was at this conference in Vienna  Chenchang Zhu. And, she
explained what's going on. So now I'm busily reading her papers:
12) HsianHua Tseng and Chenchang Zhu, Integrating Lie algebroids
via stacks, available as arXiv:math/0405003.
13) Chenchang Zhu, Lie ngroupoids and stacky Lie groupoids,
available as arXiv:math/0609420.
14) Chenchang Zhu, Lie II theorem for Lie algebroids via stacky
Lie groupoids, available as arXiv:math/0701024.
(Lie's 2nd theorem says that all Lie algebra homomorphisms
integrate to give homomorphisms between the corresponding
simplyconnected Lie groups.)
I'm optimistic that the patterns will be very beautiful when we
fully understand them. In particular, problems also arise
when trying to integrate Lie nalgebras to get Lie ngroups, but
a lot of progress has been made on these problems:
15) Ezra Getzler, Lie theory for nilpotent Linfinity algebras,
available as arXiv:math/0404003.
16) Andre Henriques, Integrating Linfinity algebras, available
as arXiv:math/0603563.
The really wonderful part is that there's already a functioning
theory of Lie nalgebroids, carefully disguised under the name of
"NQmanifolds of degree n". For a great introduction to these,
see section 2 of this paper:
17) Dmitry Roytenberg, On the structure of graded symplectic
supermanifolds and Courant algebroids, in Quantization, Poisson
Brackets and Beyond, ed. Theodore Voronov, Contemp. Math. 315,
AMS, Providence, Rhode Island, 2002. Also available as
arxiv:math.SG/0203110.
Using these, people are already busy extending the ideas of
classical mechanics across the top row of the periodic table!
The details are currently rather baroque. The best way to
see the big picture, I think, is to simultaneously read the
above papers by Pavol Severa and Dmitry Roytenberg. For example,
Roytenberg's paper proves that:
symplectic NQmanifolds of degree 0 = symplectic manifolds
symplectic NQmanifolds of degree 1 = Poisson manifolds
symplectic NQmanifolds of degree 2 = Courant algebroids
If we follow his advice and define Lie nalgebroids to be
NQmanifolds of degree n, we can express this by saying:
symplectic Lie 0algebroids = symplectic manifolds
symplectic Lie 1algebroids = Poisson manifolds
symplectic Lie 2algebroids = Courant algebroids
And ultimately, Lie nalgebroids should be just a technical
tool for studying Lie ngroupoids  modulo the tricky problems with
the generalizations of Lie's 2nd theorem, mentioned above.
Though I met both Roytenberg and Severa in Vienna, I was just
beginning to grasp the basics of NQmanifolds, Courant algebroids
and the like, so I couldn't take full advantage of this opportunity.
I will need to pester them some other time. In fact, I was stuggling
to cope with the fact that everything I just mentioned is just part
of an even bigger story....
This bigger story involves BatalinVilkovisky quantization,
Poisson sigma models, the proof by Kontsevich that every Poisson
manifold admits a deformation quantization, its interpretation
by Cattaneo and Felder in the language of 2d TQFTs, and its
generalization by Hofman and Park to the quantization of Courant
algebroids using 3d TQFTs... which should itself be the tip of a
bigger iceberg. To quantize symplectic Lie nalgebroids, it seems
we need to use (n+1)dimensional TQFTs! There are some truly
mindboggling ideas afoot here, which will turn out to be quite
simple when properly understood. For a taste of the underlying
simplicity, try this:
18) Urs Schreiber, That shift in dimension,
http://golem.ph.utexas.edu/category/2007/08/john_baez_and_i_spent.html
But, I'd better learn more before trying to explain these things.
Now, let me return to the Tale of Groupoidification! When I left
off, I was about to discuss an example: Hecke operators in the
special case of symmetric groups. But, one reader expressed
unease with what I'd done so far, saying it was too informal and
handwavy to understand.
So, this Week I'll fill in some details about "degroupoidification" 
the process that sends groupoids to vector spaces and spans of
groupoids to linear operators.
How does this work? For starters, each groupoid X gives a vector
space [X] whose basis consists of isomorphism classes of objects
of X.
Given an object x in X, let's write its isomorphism class
as [x]. So: x in X gives [x] in [X].
Next, each span of groupoids
S
/ \
/ \
/ \
v v
X Y
gives a linear operator
[S]: [X] > [Y]
Note: this operator [S] depends on the whole span, not just the
groupoid S sitting on top. So, I'm abusing notation here.
More importantly: how do we get this operator [S]? The recipe is
simple, but I think you'll profit much more by seeing where the
recipe comes from.
To figure out how it should work, we insist that degroupoidification
be something like a functor. In other words, it should get along
well with composition:
[TS] = [T] [S]
and identities:
[1_X] = 1_[X]
(Warning: today, just to confuse you, I'll write composition in
the oldfashioned backwards way, where doing S and then T is
denoted TS.)
How do we compose spans of groupoids? We do it using a "weak
pullback". In other words, given a composable pair of spans:
S T
/ \ / \
f/ \g h/ \i
/ \ / \
v v v v
X Y Z
we form the weak pullback in the middle, like this:
TS
/ \
j/ \k
/ \
v v
S T
/ \ / \
f/ \g h/ \i
/ \ / \
v v v v
X Y Z
Then, we compose the arrows along the sides to get a big span
from X to Z:
TS
/ \
/ \
/ \
fj / \ ik
/ \
/ \
/ \
/ \
v v
X Z
Never heard of "weak pullbacks"? Okay: I'll tell you what an
object in the weak pullback TS is. It's an object t in T and an
object s in S, together with an isomorphism between their images in Y.
If we were doing the ordinary pullback, we'd demand that these
images be *equal*. But that would be evil! Since t and s are living
in groupoids, we should only demand that their images be *isomorphic*
in a specified way.
(Exercise: figure out the morphisms in the weak pullback. Figure
out and prove the universal property of the weak pullback.)
So, how should we take a span of groupoids
S
/ \
/ \
/ \
v v
X Y
and turn it into a linear operator
[S]: [X] > [Y] ?
We just need to know what this operator does to a bunch of
vectors in [X]. How do we describe vectors in [X]?
I already said how to get a basis vector [x] in [X] from any object
x in X. But, that's not enough for what we're doing now, since a
linear operator doesn't usually send basis vectors to *basis*
vectors. So, we need to generalize this idea.
An object x in X is the same as a functor from 1 to X:
1

p

v
X
where 1 is the groupoid with one object and one morphism. So,
let's generalize this and figure out how *any* functor from *any*
finite groupoid V to X:
V

p

v
X
picks out a vector in [X]. Again, by abuse of notation we'll
call this vector [V], even though it also depends on p.
First suppose V is a finite set, thought of as a groupoid with
only identity morphisms. Then to define [V], we just go through
all the points of V, figure out what p maps them to  some bunch
of objects x in X  and add up the corresponding basis vectors
[x] in [X].
I hope you see how pathetically simple this idea is! It's especially
familiar when V and X are both sets. Here's what it looks like then:

V  o 
 o o 
  o o o o 
p  o o o o o 
 
v

X  o o o o o o 

I've drawn the elements of V and X as little circles, and shown how
each elements in X has a bunch of elements of V sitting over it. When
degroupoidify this to get a vector in the vector space [X], we get:
[V] = (1, 4, 3, 2, 0, 2)
This vector is just a list of numbers saying how many points of V
are sitting over each point of X!
Now we just need to generalize a bit further, to cover the case
where V is a groupoid:
V

p

v
X
Sitting over each object x in X we have its "essential preimage",
which is a groupoid. To get the vector [V], we add up basis
vectors [x] in [X], one for each isomorphism class of objects
in X, multiplied by the "cardinalities" of their essential preimages.
Now you probably have two questions:
A) Given a functor p: V > X between groupoids and an object
x in X, what's the "essential preimage" of x?
and
B) what's the "cardinality" of a groupoid?
Here are the answers:
A) An object in the essential preimage of x is an object
v in V equipped with an isomorphism from p(v) to x.
(Exercise: define the morphisms in the essential preimage.
Figure out and prove the universal property of the essential
preimage. Hint: the essential preimage is a special case of a
weak pullback!)
B) To compute the cardinality of a groupoid, we pick one object
from each isomorphism class, count its automorphisms, take the
*reciprocal* of this number, and add these numbers up.
(Exercise: check that the cardinality of the groupoid of finite
sets is e = 2.718281828... If you get stuck, read "week147".)
Also: define the morphisms in the essential preimage. Figure out
and prove the universal property of the essential preimage. Hint:
the essential preimage is a special case of a weak pullback!)
Okay. Now in principle you know how any groupoid over X, say
V



v
X
determines a vector [V] in [X]. You have to work some examples
to get a feel for it, but I want to get to the punchline. We're
unpeeling an onion here, and we're almost down to the core, where
you see there's nothing inside and wonder why you were crying so much.
So, let's finally figure out how a span of groupoids
S
/ \
/ \
/ \
v v
X Y
gives a linear operator
[S]: [X] > [Y]
It's enough to know what this operator does to vectors
coming from groupoids over X:
V



v
X
And, the trick is to notice that such a diagram is the same as
a silly span like this:
V
/ \
/ \
/ \
v v
1 X
1 is the groupoid with one object and one morphism, so there's
only one choice of the left leg here!
So here's what we do. To apply the operator [S] coming from
the span
S
/ \
/ \
/ \
v v
X Y
to the vector [V] corresponding to the silly span
V
/ \
/ \
/ \
v v
1 X
we just compose these spans, and get a silly span
SV
/ \
/ \
/ \
v v
1 Y
which picks out a vector [SV] in [Y]. Then, we define
[S] [V] = [SV]
Slick, eh? Of course you need to check that [S] is welldefined.
Given that, it's trivial to prove that [] gets along with
composition of spans:
[TS] = [T] [S]
At least, it's trivial once you know that composition of spans is
associative up to equivalence, and equivalent spans give the same
operator! But your friendly neighborhood category theorist can check
such facts in a jiffy, so let's just take them for granted. Then
the proof goes like this. We have:
[TS] [V] = [(TS)V] by definition
= [T(SV)] by those facts I just mentioned
= [T] [SV] by definition
= [T] [S] [V] by definition
Since this is true for all [V], we conclude
[TS] = [T] [S]
Voila!
By the way, if "week47" doesn't satisfy your hunger for information on
groupoid cardinality, try this:
19) John Baez and James Dolan, From finite sets to Feynman diagrams, in
Mathematics Unlimited  2001 and Beyond, vol. 1, eds. Bjorn Engquist
and Wilfried Schmid, Springer, Berlin, 2001, pp. 2950. Also
available as math.QA/0004133.
For more on turning spans of groupoids into linear operators, and
composing spans via weak pullback, try these:
20) Jeffrey Morton, Categorified algebra and quantum mechanics,
TAC 16 (2006), 785854. Available at
http://www.emis.de/journals/TAC/volumes/16/29/1629abs.html;
also available as math.QA/0601458.
21) Simon Byrne, On Groupoids and Stuff, honors thesis, Macquarie
University, 2005, available at
http://www.maths.mq.edu.au/~street/ByrneHons.pdf and
http://math.ucr.edu/home/baez/qgspring2004/ByrneHons.pdf
For a more leisurely exposition, with a big emphasis on applications
to combinatorics and the quantum mechanics of the harmonic oscillator,
try:
22) John Baez and Derek Wise, Quantization and Categorification,
Quantum Gravity Seminar lecture notes, available at:
http://math.ucr.edu/home/baez/qgfall2003/
http://math.ucr.edu/home/baez/qgwinter2004/
http://math.ucr.edu/home/baez/qgspring2004/
Finally, a technical note. Why did I say the degroupoidification
process was "something like" a functor? It's because spans of
groupoids don't want to be a category!
Already spans of sets don't naturally form a category. They form a
weak 2category! Since pullbacks are only defined up to canonical
isomorphism, composition of spans of sets is only associative
up to isomomorphism... but luckily, this "associator" isomorphism
satisfies the "pentagon identity" and all that jazz, so we get a
weak 2category, or bicategory.
Similarly, spans of groupoids form a weak 3category. Weak pullbacks
are only defined up to canonical equivalence, so composition of spans
of groupoids are associative up to equivalence... but luckily this
"associator" equivalence satisfies the pentagon identity up to an
isomorphism, and this "pentagonator" isomomorphism satisfies a
coherence law of its own, governed by the 3d Stasheff polytope.
So, we're fairly high in the ladder of ncategories. But, if we
want a mere category, we can take groupoids and *equivalence classes*
of spans. Then, degroupoidification gives a functor
[]: [finite groupoids, spans] > [vector spaces, linear maps]
That's the fact whose proof I tried to sketch here.
While I'm talking about annoying technicalities, note we need
some sort of finiteness assumption on our spans of groupoids
to be sure all the necessary sums converge. If we go allout
and restrict to spans where all groupoids involved are finite,
we'll be very safe. The cardinality of a finite groupoid is a
nonnegative rational number, so we can take our vector spaces to
be defined over the rational numbers.
But, it's also fun to consider "tame" groupoids, as defined in that
paper I wrote with Jim Dolan. These have cardinalities that can
be irrational numbers, like e. So, in this case we should use
vector spaces over the real numbers  or complex numbers, but that's
overkill.
Finding a class of groupoids or other entities whose cardinalities
are complex would be very nice, to push the whole groupoidification
program further into the complex world. In the above paper by
Jeff Morton, he uses sets over U(1), but that's probably not the
last word.

Quote of the Week:
Viewed superficially, mathematics is the result of centuries of effort
by thousands of largely unconnected individuals scattered across
continents, centuries and millennia. However the internal logic of
its development much more closely resembles the work of a single
intellect developing its thought in a continuous and systematic way 
much as in an orchestra playing a symphony written by some composer
the theme moves from one instrument to another, so that as soon as
one performer is forced to cut short his part, it is taken up by
another player, who continues with due attention to the score.
 Igor Shafarevich

Addendum: Urs Schreiber wrote:
You write:
we'll call this vector [V]
Wouldn't it be nicer to call this vector [v]? And in fact to call
V

p

v
X
instead
v

p

v
X
Seems to me that would guide the eye a little better.
Actually, thinking about it, what really deserves to be called v is
the morphism p.
I replied:
I guess a lowercase v would be nicer, so we could write Sv as the
result of applying the span of groupoids S to v. Then we could write
[Sv] = [S] [v]
It's also true that what really matters is the morphism p: v > X,
not v itself  but this notation gets a bit tricky when we apply it
to a span, or matrix, which has one object and *two* morphisms.
Probably Jim Dolan's approach is best: we should think of each morphism
as an "index", in a categorified version of Penrose's abstract
index notation. Then a groupoid over X:
v

i

v
X
can be written as a vector with one index:
v_i
Similarly, a span of groupoids:
S
/ \
i/ \j
/ \
v v
X Y
can be written as a matrix with two indices:
S_{ij}
And so on for higherrank tensors, as David Corfield pointed out.
Summing over repeated indices is then our notation for taking
weak pullbacks! And we don't need to distinguish between upper
and lower indices, due to the sneaky properties of groupoidification.
More abstractly, we can think of v as an object in [groupoids over X]
and drop the index. Similarly, we can think of S as an object in
[groupoids over X x Y] and drop the indices.
So, we have both the usual mathematicians' notation and the usual
physicists' notation available to us! Everything will look quite
ordinary, but it's all been groupoidified!
For more discussion, go to the nCategory Cafe:
http://golem.ph.utexas.edu/category/2007/08/this_weeks_finds_in_mathematic_17.html

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html