* week257
@ 2007-10-16 0:07 John Baez
0 siblings, 0 replies; 2+ messages in thread
From: John Baez @ 2007-10-16 0:07 UTC (permalink / raw)
To: categories
Dear Categorists -
I made some mistakes in my account of Cheng's work, saying "monad
on a category" at some points where I should have said "monad in
a 2-category". Here's a fixed version:
Street noted that we can talk about monads, not just in the
2-category of categories, but in any 2-category. I actually
explained monads at this level of generality back in "week89".
Indeed, for any 2-category C, there's a 2-category Mnd(C) of
monads in C.
And, he noted that a monad in Mnd(C) is a pair of monads in C
related by a distributive law!
That's already mindbogglingly beautiful. According to Eugenia,
it's practically the last sentence of Street's paper. But in
her new work:
18) Eugenia Cheng, Iterated distributive laws, available as
arXiv:0710.1120.
she goes a bit further: she considers monads in Mnd(Mnd(C)),
and so on. Here's the punchline, at least for today: she shows
that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related
by distributive laws satisfying the Yang-Baxter equation:
\F G/ |H F| G\ /H
\ / | | \ /
/ | | /
/ \ | | / \
/ \ | \ / \
| \ / \ / |
| / = / |
| / \ / \ |
| / \ / \ |
\ / | | \ /
\ / | | \ /
/ | | /
/ \ | | / \
/H \G |F H| G/ \F
This is also just what you need to make the composite FGH
into a monad!
By the way, the pathetic piece of ASCII art above is lifted
from "week1", where I first explained the Yang-Baxter equation.
That was back in 1993. So, it's only taken me 14 years to learn
that you can derive this equation from considering monads in
the category of monads in the category of monads in a 2-category.
Also, I should have given a reference to earlier work on Gelfand
duality in a topos:
Bernhard Banachewski and Christopher J. Mulvey, A globalisation
of the Gelfand duality theorem, Ann. Pure Appl. Logic 137 (2006),
62-103. Also available at
http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf
They show that any commutative C*-algebra A in a Grothendieck topos is
canonically isomorphic to the C*-algebra of continuous complex functions
on the compact, completely regular locale that is its maximal spectrum
(that is, the space of homomorphisms f: A -> C). Conversely, they show
any compact completely regular locale X gives a commutative C*-algebra
consisting of continuous complex functions on X.
^ permalink raw reply [flat|nested] 2+ messages in thread
* week257
@ 2007-10-15 3:07 John Baez
0 siblings, 0 replies; 2+ messages in thread
From: John Baez @ 2007-10-15 3:07 UTC (permalink / raw)
To: categories
Dear Categorists -
The latest issue of This Week's Finds contains enough category theory
that I felt like sharing it here, especially since it contains links
to the Catsters videos and my seminar with Jim Dolan on geometric
representation theory. This seminar will ultimately discuss=20
"groupoidification", an idea also discussed in This Week's Finds
starting in "week247".
The web version has some pretty pictures.
Best,
jb
...................................................................
Also available as http://math.ucr.edu/home/baez/week257.html
October 14, 2007
This Week's Finds in Mathematical Physics (Week 257)
John Baez
Time flies! This week I'll finally finish saying what I did on=20
my summer vacation. After my trip to Oslo I stayed in London,=20
or more precisely Greenwich. While there, I talked with some good=20
mathematicians and physicists: in particular, Minhyong Kim, Ray=20
Streater, Andreas Doering and Chris Isham. I also went to a=20
topology conference in Sheffield... and Eugenia Cheng explained
some cool stuff on the train ride there. I want to tell you about=20
all this before I forget.
Also, the Tale of Groupoidification has taken a shocking new
turn: it's now becoming available as a series of *videos*.
But first, some miscellaneous fun stuff on math and astronomy. =20
Math: if you haven't seen a sphere turn inside out, you've got=20
to watch this classic movie, now available for free online:
1) The Geometry Center, Outside in,
http://video.google.com/videoplay?docid=3D-6626464599825291409
Astronomy: did you ever wonder where dust comes from? I'm=20
not talking about dust bunnies under your bed - I'm talking=20
about the dust cluttering our galaxy, which eventually clumps=20
together to form planets and... you and me!
These days most dust comes from aging stars called "asymptotic giant
branch" stars. The sun will eventually become one of these. The
story goes like this: first it'll keep burning until the hydrogen in
its core is exhausted. Then it'll cool and become a red giant.
Eventually helium at the core will ignite, and the Sun will shrink=20
and heat up again... but its core will then become cluttered with even=20
heavier elements, so it'll cool and expand once more, moving onto the
"asymptotic giant branch". At this point it'll have a layered
structure: heavier elements near the bottom, then a layer of helium,
then hydrogen on the top.
(A similar fate awaits any star between 0.6 and 10 solar masses,
though the details depend on the mass. For the more dramatic
fate of heavier stars, see "week204".)
This layered structure is unstable, so asymptotic giant branch=20
stars pulse every 10 to 100 thousand years or so. And, they=20
puff out dust! Stellar wind then blows this dust out into space. =20
A great example is the Red Rectangle:
2) Rungs of the Red Rectangle, Astronomy picture of the day,=20
May 13, 2004, http://apod.nasa.gov/apod/ap040513.html
Here two stars 2300 light years from us are spinning around
each other while pumping out a huge torus of icy dust grains and
hydrocarbon molecules. It's not really shaped like a rectangle=20
or X - it just looks that way. The scene is about 1/3 of a light=20
year across.
Ciska Markwick-Kemper is an expert on dust. She's an astrophysicist
at the University of Manchester. Together with some coauthors, she
wrote a paper about the Red Rectangle:
3) F. Markwick-Kemper, J. D. Green, E. Peeters, Spitzer=20
detections of new dust components in the outflow of the Red=20
Rectangle, Astrophys. J. 628 (2005) L119-L122. Also available
as arXiv:astro-ph/0506473.
They used the Spitzer Space Telescope - an infrared telescope on=20
a satellite in earth orbit - to find evidence of magnesium and=20
iron oxides in this dust cloud. =20
But, what made dust in the early Universe? It took about a
billion years after the Big Bang for asymptotic giant branch stars
to form. But we know there was a lot of dust even before then!
We can see it in distant galaxies lit up by enormous black holes=20
called "quasars", which pump out vast amounts of radiation as=20
stuff falls into them. =20
Markwick-Kemper and coauthors have also tackled that question:
4) F. Markwick-Kemper, S. C. Gallagher, D. C. Hines and J. Bouwman,=20
Dust in the wind: crystalline silicates, corundum and periclase in=20
PG 2112+059, Astrophys. J. 668 (2007), L107-L110. Also available
as arXiv:0710.2225.
They used spectroscopy to identify various kinds of dust in=20
a distant galaxy: a magnesium silicate that geologists call=20
"forsterite", a magnesium oxide called "periclase", and aluminum
oxide, otherwise known as "corundum" - you may have seen it on=20
sandpaper.
And, they hypothesize that these dust grains were formed in the
hot wind emanating from the quasar at this galaxy's core!
So, besides being made of star dust, as in the Joni Mitchell
song, you also may contain a bit of black hole dust.=20
Okay - now that we've got that settled, on to London!
Minhyong Kim is a friend I met back in 1986 when he was a grad=20
student at Yale. After dabbling in conformal field theory, he
became a student of Serge Lang and went into number theory. He=20
recently moved to England and started teaching at University=20
College, London. I met him there this summer, in front of the=20
philosopher Jeremy Bentham, who had himself mummified and stuck
in a wooden cabinet near the school's entrance.
If you're not into number theory, maybe you should read this:
5) Minhyong Kim, Why everyone should know number theory,
available at http://www.ucl.ac.uk/~ucahmki/numbers.pdf
Personally I never liked the subject until I realized it was
a form of *geometry*. For example, when we take an equation like
this
x^2 + y^3 =3D 1
and look at the real solutions, we get a curve in the plane -=20
a "real curve". If we look at the complex solutions, we get
something bigger. People call it a "complex curve", because=20
it's analogous to a real curve. But topologically, it's=20
2-dimensional. This will be important in a few minutes, so=20
don't forget it!
If we use polynomial equations with more variables, we get=20
higher-dimensional shapes called "algebraic varieties" - either=20
real or complex. Either way, we can study these shapes using=20
geometry and topology. =20
But in number theory, we might study the solutions of these=20
equations in some other number system - for example in Z/p,=20
meaning the integers modulo some prime p. At first glance there's=20
no geometry involved anymore. After all, there's just a *finite=20
set* of solutions! However, algebraic geometers have figured=20
out how to apply ideas from geometry and topology, mimicking=20
tricks that work for the real and complex numbers. =20
All this is very fun and mind-blowing - especially when we reach
Grothendieck's idea of "etale topology", developed around 1958. =20
This is a way of studying "holes" in things like algebraic=20
varieties over finite fields. Amazingly, it gives results that=20
nicely match the results we get for the corresponding complex
algebraic varieties! That's part of what the "Weil conjectures"
say.
You can learn the details here:
6) J. S. Milne, Lectures on Etale Cohomology, available at
http://www.jmilne.org/math/CourseNotes/math732.html
Anyway, I quizzed about Minhyong about one of the big mysteries
that's been puzzling me lately. I want to know why the integers=20
resemble a 3-dimensional space - and how prime numbers are like=20
"knots" in this space! =20
Let me try to explain this in a very sketchy way, without getting=20
into any technical details. I'll still make mistakes... but this=20
stuff is just too cool to keep secret - so if the experts don't=20
explain it, nonexperts like me have to try.
You can think of Z/p as giving a very simple sort of curve. =20
Naively you could imagine it as shaped like a ring, for example=20
the integers mod 7 here:
0
6 1
=20
5 2
4 3
=20
But now it's better to think of Z/p as a "line". After=20
all, a line is defined by one variable and no equations. Here=20
we have one variable in Z/p. =20
But remember: a curve defined in a field like Z/p acts a lot=20
like a complex curve. And, a complex curve is topologically=20
2-dimensional! =20
So, the "line" associated to Z/p seems 2-dimensional from the=20
viewpoint of etale topology. In other words, it's really more=20
like a "plane" - just like the complex numbers are topologically a=20
plane.
This is true for each prime p. But the integers, Z, are more=20
complicated than any of these Z/p's. To be precise, we have maps
Z -> Z/p
for each p. So, if we think of Z as a kind of space, it's a big=20
space that contains all the "planes" corresponding to the Z/p's. =20
So, it's 3-dimensonal! =20
In short: from the viewpoint of etale topology, the integers have=20
one dimension that says which prime you're at, and two more coming=20
from the plane-like nature of each individual Z/p. =20
Naively you might imagine a stack of planes, one for each prime. =20
But that's a very crude picture, and it misses a crucial fact: the=20
primes get "tangled up" with each other. In fact, each "plane" has=20
a specially nice circle in it, and these circles are *linked*. =20
I've been fascinated by this ever since I heard about it, but I
got even more interested when I saw a draft of a paper by=20
Kapranov and some coauthor. I got it from Thomas Riepe, who got
it from Yuri Manin. I don't have it right here with me, so I'll
add a reference later... but I don't think it's available yet,
so the reference won't do you much good anyway.
In this paper, the authors explain how the "Legendre symbol" of=20
primes is analogous to the "linking number" of knots.
The Legendre symbol depends on two primes: it's 1 or -1 depending=20
on whether or not the first is a square modulo the second. The=20
linking number depends on two knots: it says how many times the=20
first winds around the second.
The linking number stays the same when you switch the two knots. =20
The Legendre symbol has a subtler symmetry when you switch the=20
two primes: this symmetry is called "quadratic reciprocity", and=20
it has lots of proofs, starting with a bunch by Gauss - all a bit=20
tricky. =20
I'd feel very happy if I truly understood why quadratic reciprocity=20
reduces to the symmetry of the linking number when we think of=20
primes as analogous to knots. Unfortunately, I'll need to think a=20
lot more before I really get the idea. I got into number theory=20
late in life, so I'm pretty slow at it. =20
This paper studies subtler ways in which primes can be "linked":
7) Masanori Morishita, Milnor invariants and Massey products for=20
prime numbers, Compositio Mathematica 140 (2004), 69-83.
You may know the Borromean rings, a design where no two rings are
linked in isolation, but all three are when taken together. Here=20
the linking numbers are zero, but the linking can be detected by=20
something called the "Massey triple product". Morishita=20
generalizes this to primes!
But I want to understand the basics...
The secret 3-dimensional nature of the integers and certain other=20
"rings of algebraic integers" seems to go back at least to the work=20
of Artin and Verdier:
8) Michael Artin and Jean-Louis Verdier, Seminar on etale cohomology=20
of number fields, Woods Hole, 1964.=20
You can see it clearly here, starting in section 2:
9) Barry Mazur, Notes on the etale cohomology of number fields,
Annales Scientifiques de l'Ecole Normale Superieure Ser. 4,=20
6 (1973), 521-552. Also available at
http://www.numdam.org/numdam-bin/fitem?id=3DASENS_1973_4_6_4_521_0
By now, a big "dictionary" relating knots to primes has been=20
developed by Kapranov, Mazur, Morishita, and Reznikov. This=20
seems like a readable introduction:
10) Adam S. Sikora, Analogies between group actions on 3-manifolds
and number fields, available as arXiv:math/0107210.
I need to study it. These might also be good - I haven't looked
at them yet:
11) Masanori Morishita, On certain analogies between knots and=20
primes, J. Reine Angew. Math. 550 (2002), 141-167.
Masanori Morishita, On analogies between knots and primes,=20
Sugaku 58 (2006), 40-63.
After giving a talk on 2-Hilbert spaces at University College, I went
to dinner with Minhyong and some folks including Ray Streater. Ray
Streater and Arthur Wightman wrote the book "PCT, Spin, Statistics and
All That". Like almost every mathematician who has seriously tried to
understand quantum field theory, I've learned a lot from this book.
So, it was fun meeting Streater, talking with him - and finding out
he'd once been made an honorary colonel of the US Army to get a free
plane trip to the Rochester Conference! This was a big important
particle physics conference, back in the good old days.
He also described Geoffrey Chew's Rochester conference talk on the=20
analytic S-matrix, given at the height of the bootstrap theory fad.=20
Wightman asked Chew: why assume from the start that the S-matrix was=20
analytic? Why not try to derive it from simpler principles? Chew=20
replied that "everything in physics is smooth". Wightman asked about
smooth functions that aren't analytic. Chew thought a moment and=20
replied that there weren't any.
Ha-ha-ha...
What's the joke? Well, first of all, Wightman had already succeeded
in deriving the analyticity of the S-matrix from simpler principles.=20
Second, any good mathematician - but not necessarily every physicist,=20
like Chew - will know examples of smooth functions that aren't=20
analytic.=20
Anyway, Streater has just finished an interesting book on "lost=20
causes" in physics: ideas that sounded good, but never panned out. =20
Of course it's hard to know when a cause is truly lost. But a=20
good pragmatic definition of a lost cause in physics is a topic=20
that shouldn't be given as a thesis problem. =20
So, if you're a physics grad student and some professor wants you to=20
work on hidden variable theories, or octonionic quantum mechanics,=20
or deriving laws of physics from Fisher information, you'd better=20
read this:
11) Ray F. Streater, Lost Causes in and Beyond Physics, Springer=20
Verlag, Berlin, 2007.
(I like octonions - but I agree with Streater about not inflicting=20
them on physics grad students! Even though all my students are in=20
the math department, I still wouldn't want them working mainly on=20
something like that. There's a lot of more general, clearly useful=20
stuff that students should learn.)=20
I also spoke to Andreas Doering and Chris Isham about their work=20
on topos theory and quantum physics. Andreas Doering lives near
Greenwich, while Isham lives across the Thames in London proper.
So, I talked to Doering a couple times, and once we visited Isham
at his house.
I mainly mention this because Isham is one of the gurus of quantum
gravity, profoundly interested in philosophy... so I was surprised,
at the end of our talk, when he showed me into a room with a huge=20
rack of computers hooked up to a bank of about 8 video monitors,
and controls reminiscent of an airplane cockpit.
It turned out to be his homemade flight simulator! He's been a=20
hobbyist electrical engineer for years - the kind of guy who=20
loves nothing more than a soldering iron in his hand. He'd just=20
gotten a big 750-watt power supply, since he'd blown out his
previous one. =20
Anyway, he and Doering have just come out with a series of papers:
11) Andreas Doering and Christopher Isham, A topos foundation=20
for theories of physics: I. Formal languages for physics,=20
available as arXiv:quant-ph/0703060.
II. Daseinisation and the liberation of quantum theory,=20
available as arXiv:quant-ph/0703062.
III. The representation of physical quantities with arrows,
available as arXiv:quant-ph/0703064.
IV. Categories of systems, available as arXiv:quant-ph/0703066.
Though they probably don't think of it this way, you can think=20
of their work as making precise Bohr's ideas on seeing the quantum
world through classical eyes. Instead of talking about all
observables at once, they consider collections of observables that
you can measure simultaneously without the uncertainty principle
kicking in. These collections are called "commutative subalgebras".=20
You can think of a commutative subalgebra as a classical snapshot
of the full quantum reality. Each snapshot only shows part of the
reality. One might show an electron's position; another might show
its momentum.
Some commutative subalgebras contain others, just like some open=20
sets of a topological space contain others. The analogy is a good=20
one, except there's no one commutative subalgebra that contains
*all* the others. =20
Topos theory is a kind of "local" version of logic, but where the=20
concept of locality goes way beyond the ordinary notion from=20
topology. In topology, we say a property makes sense "locally"=20
if it makes sense for points in some particular open set.
In the Doering-Isham setup, a property makes sense "locally" if
it makes sense "within a particular classical snapshot of reality" -
that is, relative to a particular commutative subalgebra.
(Speaking of topology and its generalizations, this work on topoi and=20
physics is related to the "etale topology" idea I mentioned a while=20
back - but technically it's much simpler. The etale topology lets
you define a topos of "sheaves" on a certain category. The=20
Doering-Isham work just uses the topos of "presheaves" on the poset
of commutative subalgebras. Trust me - while this may sound scary,=20
it's much easier.) =20
Doering and Isham set up a whole program for doing physics=20
"within a topos", based on existing ideas on how to do math in=20
a topos. You can do vast amounts of math inside any topos just=20
as if you were in the ordinary world of set theory - but using=20
intuitionistic logic instead of classical logic. Intuitionistic
logic denies the principle of excluded middle, namely:
"For any statement P, either P is true or not(P) is true."
In Doering and Isham's setup, if you pick a commutative subalgebra=20
that contains the position of an electron as one of its observables,
it can't contain the electron's momentum. That's because these
observables don't commute: you can't measure them both simultaneously.
So, working "locally" - that is, relative to this particular=20
subalgebra - the statement
P =3D "the momentum of the electron is zero"
is neither true nor false! It's just not defined.
Their work has inspired this very nice paper:
12) Chris Heunen and Bas Spitters, A topos for algebraic quantum
theory, available as arXiv:0709.4364.
so let me explain that too.
I said you can do a lot of math inside a topos. In particular,=20
you can define an algebra of observables - or technically, a
"C*-algebra".
By the Isham-Doering work I just sketched, any C*-algebra of=20
observables gives a topos. Heunen and Spitters show that=20
the original C*-algebra gives rise to a commutative
C*-algebra in this topos, even if the original one was=20
noncommutative!
That actually makes sense, since in this setup, each "local view"=20
of the full quantum reality is classical. What's really neat is=20
that the Gelfand-Naimark theorem, saying commutative C*-algebras=20
are always algebras of continuous functions on compact Hausdorff=20
spaces, can be generalized to work within any topos. So, we get=20
a space *in our topos* such that observables of the C*-algebra=20
*in the topos* are just functions on this space. =20
I know this sounds technical if you're not into this stuff. But
it's really quite wonderful. It basically means this: using topos=20
logic, we can talk about a classical space of states for a quantum=20
system! However, this space typically has "no global points". In=20
other words, there's no overall classical reality that matches all=20
the classical snapshots. =20
As you can probably tell, category theory is gradually seeping
into this post, though I've been doing my best to keep it
hidden. Now I want to say what Eugenia Cheng explained on=20
that train to Sheffield. But at this point, I'll break down and
assume you know some category theory - for example, monads.
If you don't know about monads, never fear! I defined them in=20
"week89", and studied them using string diagrams in "week92".=20
Even better, Eugenia Cheng and Simon Willerton have formed a=20
little group called the Catsters - and under this name, they've=20
put some videos about monads and string diagrams onto YouTube! =20
This is a really great new use of technology. So, you should=20
also watch these:
14) The Catsters, Monads,=20
http://youtube.com/view_play_list?p=3D0E91279846EC843E
The Catsters, Adjunctions,=20
http://youtube.com/view_play_list?p=3D54B49729E5102248
The Catsters, String diagrams, monads and adjunctions,
http://youtube.com/view_play_list?p=3D50ABC4792BD0A086
A very famous monad is the "free abelian group" monad
F: Set -> Set
which eats any set X and spits out the free abelian group on X,=20
say F(X). A guy in F(X) is just a formal linear combination
of guys in X, with integer coefficients.
Another famous monad is the "free monoid" monad=20
G: Set -> Set
This eats any set X and spits out the free monoid on X, namely=20
G(X). A guy in G(X) is just a formal product of guys in X.
Now, there's yet another famous monad, called the "free=20
ring" monad, which eats any set X and spits out the free ring on
this set. But, it's easy to see that this is just F(G(X))!
After all, F(G(X)) consists of formal linear combinations of
formal products of guys in X. But that's precisely what you find
in the free ring on X. =20
But why is FG a monad? There's more to a monad than just a=20
functor. A monad is really a kind of *monoid* in the world of
functors from our category (here Set) to itself. In particular,=20
since F is a monad, it comes with a natural transformation called
a "multiplication":
m: FF =3D> F
which sends formal linear combinations of formal linear combinations
to formal linear combinations, in the obvious way. Similarly,
since G is a monad, it comes with a natural transformation
n: GG =3D> G
sending formal products of formal products to formal products.
But how does FG get to be a monad? For this, we need some=20
natural transformation from FGFG to FG!
There's an obvious thing to try, namely
mn=20
FGFG =3D=3D=3D=3D=3D=3D> FFGG =3D=3D=3D=3D=3D=3D> FG
where in the first step we switch G and F somehow, and in the
second step we use m and n. But, how do we do the first step?
We need a natural transformation
d: GF =3D> FG
which sends formal products of formal linear combinations
to formal linear combinations of formal products. Such a
thing obviously exists; for example, it sends
(x + 2y)(x - 3z)=20
to
xx + 2yx - 3xz - 6yz
It's just the distributive law! =20
Quite generally, to make the composite of monads F and G=20
into a new monad FG, we need something that people call a
"distributive law", which is a natural transformation
d: GF =3D> FG
This must satisfy some equations - but you can work out
those yourself. For example, you can demand that
FdG mn=20
FGFG =3D=3D=3D=3D=3D=3D> FFGG =3D=3D=3D=3D=3D=3D> FG
make FG into a monad, and see what that requires. Besides the=20
"multiplication" in our monad, we also need the "unit", so you=20
should also think about that - I'm ignoring it here because it's
less sexy than the multiplication, but it's equally essential.
However: all this becomes more fun with string diagrams!
As the Catsters explain, and I explained in "week89", the=20
multiplication m: FF =3D> F can be drawn like this:
\ /
\ /
F\ F/
\ /
\ /
\ /
\ /
\ /
|m =20
|
|
|
|
|
F|
|
And, it has to satisfy the associative law, which says we
get the same answer either way when we multiply three things:
\ / / \ \ /
\ / / \ \ /
F\ /F F/ F\ F\ /F
\/ / \ \/
m\ / \ /m=20
\ / \ /
F\ / \ /F
\ / \ /
|m |m
| |
| =3D |
| |
| |
| |
F| F|
| |
The multiplication n: GG =3D> G looks similar to m, and it too has
to satisfy the associative law. =20
How do we draw the distributive law d: FG =3D> GF? Since it's a=20
process of switching two things, we draw it as a *braiding*:
F\ /G
\ /
/=20
/ \
G/ \F=20
I hope you see how incredibly cool this is: the good old=20
distributive law is now a *braiding*, which pushes our diagrams
into the third dimension! =20
Given this, let's draw the multiplication for our would-be
monad FG, namely=20
FdG mn=20
FGFG =3D=3D=3D=3D=3D=3D> FFGG =3D=3D=3D=3D=3D=3D> FG
It looks like this:
\ \ / /
\ \ / /
F\ G\ F/ /G
\ \ / /
\ \ / /
\ \ / /
\ / /
\ / \ /
|m |n =20
| |
| |
| |
| |
| |
F| |G
| |
Now, we want *this* multiplication to be associative! So,=20
we need to draw an equation like this:
\ / / \ \ /
\ / / \ \ /
\ / / \ \ /
\/ / \ \/
\ / \ /=20
\ / \ /
\ / \ /
\ / \ /
| |
| |
| =3D |
| |
| |
| |
| |
| |=20
but with the strands *doubled*, as above - I'm too lazy to draw=20
this here. And then we need to find some nice conditions that=20
make this associative law true. Clearly we should use the=20
associative laws for m and n, but the "braiding" - the=20
distributive law d: FG =3D> GF - also gets into the act.
I'll leave this as a pleasant exercise in string diagram=20
manipulation. If you get stuck, you can peek in the back of=20
the book:
14) Wikipedia, Distibutive law between monads,=20
http://en.wikipedia.org/wiki/Distributive_law_between_monads
The two scary commutative rectangles on this page are the=20
"nice conditions" you need. They look nicer as string=20
diagrams. One looks like this:
F\ G\ /G F\ G/ /G
\ \ / \ / /
\ |n \ / /
\ / / /
\ / =3D / \ /
/ / /
/ \ / /\=20
/ \ \ / \
/ \ \ / \
G/ \F |n \F
/ \ G| \
In words:=20
"multiply two G's and slide the result over an F" =3D
"slide both the G's over the F and then multiply them"
If the pictures were made of actual string, this would be obvious!
The other condition is very similar. I'm too lazy to draw it,
but it says=20
"multiply two F's and slide the result under a G" =3D=20
"slide both the F's under a G and then multiply them"
All this is very nice, and it goes back to a paper by Beck:
15) Jon Beck, Distributive laws, Lecture Notes in Mathematics=20
80, Springer, Berlin, pp. 119=96140.=20
This isn't what Eugenia explained to me, though - I already knew
this stuff. She started out by explaining something in a paper=20
by Street:
16) Ross Street, The formal theory of monads, J. Pure Appl. Alg.
2 (1972), 149-168.
which is reviewed at the beginning here:
17) Steve Lack and Ross Street, The formal theory of monads II,
J. Pure Appl. Alg. 175 (2002), 243-265. Also available at
http://www.maths.usyd.edu.au/u/stevel/papers/ftm2.html
(Check out the cool string diagrams near the end!) =20
Street noted that for any category C, there's a category Mnd(C)=20
whose objects are monads on C and whose morphisms are "monad
transforms": functors from C to C that make an obvious square
commute. =20
And, he noted that a monad on Mnd(C) is a pair of monads on C
related by a distributive law!
That's already mindbogglingly beautiful. According to Eugenia,
it's in the last sentence of Street's paper. But in her new work:
18) Eugenia Cheng, Iterated distributive laws, available as
arXiv:0710.1120.
she goes a bit further: she considers monads in Mnd(Mnd(C)),=20
and so on. Here's the punchline, at least for today: she shows=20
that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related=20
by distributive laws satisfying the Yang-Baxter equation:
\F G/ |H F| G\ /H
\ / | | \ /
/ | | /
/ \ | | / \
/ \ | \ / \
| \ / \ / |
| / =3D / |=20
| / \ / \ |
| / \ / \ |
\ / | | \ /
\ / | | \ /
/ | | /
/ \ | | / \
/H \G |F H| G/ \F
This is also just what you need to make the composite FGH
into a monad!
By the way, the pathetic piece of ASCII art above is lifted=20
from "week1", where I first explained the Yang-Baxter equation.
That was back in 1993. So, it's only taken me 14 years to learn
that you can derive this equation from considering monads on
the category of monads on the category of monads on a category.
You may wonder if this counts as progress - but Eugenia
studies lots of *examples* of this sort of thing, so it's far
from pointless. =20
Okay... finally, the Tale of Groupoidification. I'm a bit tired
now, so instead of telling you more of the tale, let me just say
the big news.
Starting this fall, James Dolan and I are running a seminar on
geometric representation theory, which will discuss:
Actions and representations of groups, especially symmetric groups
Hecke algebras and Hecke operators
Young diagrams
Schubert cells for flag varieties
q-deformation=20
Spans of groupoids and groupoidification
This is the Tale of Groupoidification in another guise.
Moreover, the Catsters have inspired me to make videos of this=20
seminar! You can already find some here, along with course=20
notes and blog entries where you can ask questions and talk about=20
the material:
19) John Baez and James Dolan, Geometric representation theory seminar,
http://math.ucr.edu/home/baez/qg-fall2007/
More will show up in due course. I hope you join the fun.
-----------------------------------------------------------------------
Quote of the Week:
It is a glorious feeling to discover the unity of a set of phenomena
that at first seem completely separate. - Albert Einstein
-----------------------------------------------------------------------
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 jumping-off 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
=20
http://math.ucr.edu/home/baez/this.week.html
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