- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week244.html
February 2, 2006
This Week's Finds in Mathematical Physics (Week 244)
John Baez
In January I spent a week at this workshop at the Fields Institute
in Toronto:
1) Higher Categories and Their Applications,
http://math.ucr.edu/home/baez/fields/
It was really fun - lots of people working on n-categories were
there. I'll talk about it next time. But as usual, more happens at
a fun conference than can possibly be reported. So, this time I'll
only talk about a conversation I had in a cafe before the conference
started!
But first, here's a fun way to challenge your math pals:
Q: When the first calculus textbook was written - and in what
language?
A: In 1530, in Malayalam - a south Indian language!
This book is called the Ganita Yuktibhasa, or "compendium of
astronomical rationales". It was written by Jyesthadeva, an
astronomer and mathematician from Kerala - a state on the southwest
coast of India. It summarizes and explains the work of many
researchers of the Kerala school, which flourished from the
1400's to the 1600's. But it's unique for its time, since it
contains proofs of many results.
For example, it has a proof that
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
Of course, this result isn't stated in modern notation! It's
actually stated as a poem - a recipe for the circumference of
a circle, which in translation goes something like this:
"Multiply the diameter by four. Subtract from it and add to it
alternately the quotients obtained by dividing four times the
diameter by the odd numbers 3, 5, etc."
The proof sounds nice! Jyesthadeva starts with something like this:
pi/4 = lim_{N -> infinity} (1/N) Sum_{n=1}^N 1/(1 + (n/N)^2)
In modern terms, the right-hand side is just the integral
integral_0^1 dx/(1 + x^2)
You can use geometry to see this equals pi/4. Then, as far as
I can tell, he writes
1/(1 + (n/N)^2) = 1 - (n/N)^2 + (n/N)^4 - ...
and notes that
1^k + 2^k + ... + N^k ~ N^{k+1}/(k+1)
for large N. This gives
pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...
Voila!
In fact, this result goes back to Madhava, an amazing mathematician
from Kerala who lived much earlier, from 1350 to 1425. What's even
more impressive is that Madhava also knew a formula equivalent to the
more general result
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
He used this to compute pi to 11 decimal places!
It's an interesting question whether any of the results of the
Kerala school found their way west and influenced the development
of mathematics in Europe. There's been a lot of speculation, but
nobody seems to know for sure. For more info, try these:
2) The MacTutor History of Mathematics Archive, Madhava of
Sangamagramma,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Madhava.html
3) The MacTutor History of Mathematics Archive, Jyesthadeva,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Jyesthadeva.html
4) Wikipedia, Yuktibhasa,
http://en.wikipedia.org/wiki/Yuktibhasa
Before the conference started, I spent a nice morning talking with Tom
Leinster in a cafe on Bloor Street. There's nothing like talking
about math in a nice warm cafe when it's cold outside! At some point
my former grad student Toby Bartels showed up - he'd just taken a long
Greyhound bus from Nebraska - and joined in the conversation. We
talked about this paper:
5) Tom Leinster, The Euler characteristic of a category,
available as math.CT/0610260.
Everyone know how to measure the size of a set - by its number
of elements, or "cardinality". But what's the size of a category?
That's the question this paper tackles!
Some categories are just sets in disguise: the "discrete" categories,
whose only morphisms are identity morphisms. We'd better define the
size of such a category to be the cardinality of its set of objects.
For example, the category with just one object and its identity
morphism is called 1. It looks sort of like this:
o
where I've drawn the object but not its identity morphism. Clearly,
its size should be 1.
We could also have a category with just two objects and their identity
morphisms. It looks like this:
o o
and its size should be 2.
But what about this?
o--<--->--o
Here we have a category with two objects and an invertible morphism
between them, which I've drawn as an arrow pointing both ways. Again,
I won't draw the identity morphisms.
In other words, we have two objects that are *isomorphic* - and in a
unique way. How big should this category be?
Any mathematician worth her salt knows that having two things that are
isomorphic in a unique way is just like having one: you can't do
anything more with them - or less. So, the size of this category:
o--<--->--o
should equal the size of this one:
o
namely, 1.
More technically, we say these categories are "equivalent".
We'll demand that equivalent categories have the same size.
This is a powerful principle. If we didn't insist on this,
we'd be insane.
But what about this category:
o---->----o
Now we have two objects and a morphism going just one way!
This is *not* equivalent to a discrete category, so we need
a new idea to define its size.
If we were willing to make up new kinds of numbers, we could
make up a new number for the size of this category. But let's
suppose that this is against the rules.
There's a cute way to turn any category into a space, which I
described in "week70" - and in more detail in items J and K of
"week117", back when I was giving a minicourse on homotopy theory.
If we do this to the category
o---->----o
what do we get? The unit interval, of course! It's a pretty
intuitive notion, at least in this example.
We also get the unit interval if we turn this guy
o--<--->--o
into a space. So, even though these categories aren't equivalent,
they give the same space. So, let's declare that they have the
same size - namely, 1.
In fact, let's adopt this as a new principle! We'll demand that
two categories have the same size whenever they give the same space.
Whenever categories are equivalent, they give the same space (where "the
same" means "homotopy equivalent"). So, our new principle includes our
previous principle as a special case. But, we can say more. If you like
adjoint functors, you'll enjoy this: whenever there's a pair of adjoint
functors going between two categories, they give the same space. For
example, these categories
o---->----o
and
o--<--->--o
aren't equivalent, but there's a pair of adjoint functors going between
them. (If you don't like adjoint functors, oh well - just ignore this.)
Next, what's the size of this category?
---->----
/ \
o o
\ /
---->----
This is my feeble attempt to draw a category with two objects, and two
morphisms going from the first object to the second.
If we turn this category into the space, what do we get? The circle, of
course! But what's the "size", or "cardinality", of a circle?
That's a tricky puzzle, because it's hard to know what counts as a right
answer. It turns out the right answer is zero. Why? Because the
"Euler characteristic" of the circle is zero!
As you may know, Euler lived in Konigsberg, a city with lots of islands
and bridges. In fact, he published a paper in 1736 showing that you
can't walk around Konigsberg and cross each bridge exactly once, winding
up where you started. My crazy theory is that living there also helped
him invent the concept of Euler characteristic. I have no evidence for
this, except for this apocryphal story I just made up:
Once upon a time, Euler was strolling along one of the bridges of
Konigsberg. He looked across the river, and noticed that workers
werebuilding a bridge to a small island that had previously been
unconnected to the rest. He noticed that this reduced the number of
isolated islands by one. Of course, anyone could have seen that!
But in a burst of genius, Euler went further - he realized this meant a
bridge was like a "negative island". And so, he invented the concept
of "Euler characteristic". In its simplest form, it's just the number
of islands minus the number of bridges.
For example, if you have two islands in the sea:
o o
the land has Euler characteristic 2.
If you build a bridge:
o--<--->--o
the land now has Euler characteristic 1. This makes sense, because the
land is now effectively just one island. So, a bridge acts as a
"negative island"!
But now, if you build a *second* bridge:
---->----
/ \
o o
\ /
---->----
the land has Euler characteristic 0. This is sort of weird. But, Euler
saw it was a good idea.
To understand why, you have to go further and imagine building a "bridge
between bridges" - filling in the space between the bridges with an
enormous deck:
---->----
/xxxxxxxxx\
oxxxxxxxxxxxo
\xxxxxxxxx/
---->----
This reduces the number of bridges by one. We've effectively got one
island again, though much bigger now. So, we're back to having Euler
characteristic 1.
In short, adding a "bridge between bridges" should add 1 to the Euler
characteristic. Just as a bridge counts as a negative island, a bridge
between bridges counts as a negative bridge - or an island:
-(-1) = 1.
It's all consistent, in its own weird way.
So, Euler defined the Euler characteristic to be
V - E + F
where V is the number of islands (or "vertices"), E is the number of
bridges (or "edges") and F is the number of bridges between bridges (or
"faces").
At least that's how the story goes.
By the way, you must have noticed that the number 1 looks like an
interval, while the number 0 looks like a circle. But did you notice
that the Euler characteristic of the interval is 1, and the Euler
characteristic of the circle is 0? I can never make up my mind whether
this is a coincidence or not.
Anyway, we can easily generalize the Euler characteristic to higher
dimensions, and define it as an alternating sum. And that turns out
to be important for us now, because it turns out that often when we
turn a category into a space, we get something higher-dimensional!
This shouldn't be obvious, since I haven't told you the rule for turning
a category into a space. You might think we always get something
1-dimensional, built from vertices (objects) and edges (morphisms).
But the rule is more subtle. Whenever we have 2 morphisms end to end,
like this:
f g
o--->---o--->---o
X Y Z
we can compose them and get a morphism fg going all the way from x to z.
We should draw this morphism too... so the space we get is a *triangle*:
Y
o
/x\
f /xxx\ g
/xxxxx\
X o-------o Z
fg
I haven't drawn the arrows on my morphisms, due to technical limitations
of this medium. More importantly, the triangle is filled with x's, just
like Euler's "bridge between bridges", to show that it's *solid*, not
hollow.
Simlarly, when we have 3 morphisms laid end to end we get a tetrahedron,
and so on.
Using these rules, it's not hard to find a category that gives a sphere,
or a torus, or an n-holed torus, when you turn it into a space. I'll
leave that as a puzzle.
In fact, for *any* manifold, you can find a category that gives you
that manifold when you turn it into a space! In fact we can get any
space at all this way, up to "weak homotopy equivalence" - whatever
that means. So, let's adopt a new principle: whenever our category
gives a space whose Euler characteristic is well-defined, we should
define the size of our category to be that.
I say "when it's well-defined", because it's also possible for a
category - even one with just finitely many objects and morphisms - to
give an infinite-dimensional space whose Euler characteristic is a
divergent series:
n_0 - n_1 + n_2 - n_3 + n_4 - ...
Okay. At this point it's time for me to say what Leinster actually did:
he came up with a *formula* that you can use to compute the size of a
category, without using any topology. Sometimes it gives divergent
answers - which is no shame: after all, some categories are infinitely
big. But when it converges, it satisfies all the principles I've
mentioned.
Even better, it works for a lot of categories that give spaces whose
Euler chacteristic diverges! For example, we can take any group G and
think of it as a category with one object, with the group elements as
morphisms. When we turn this category into a space, it becomes
something famous called the "classifying space" of G. This is often
an infinite-dimensional monstrosity whose Euler characteristic diverges.
But, Leinster's formula still works - and it gives
1/|G|
the reciprocal of the usual cardinality of G.
Now we're getting fractions!
For example, suppose we take G to be the group with just 2 elements,
called Z/2. If we think of it as a category, and then turn that into
a space, we get a huge thing usually called "infinite-dimensional real
projective space", or RP^infinity for short. This is built from one
vertex, one edge, one triangle, and so on. So, if we try to work out
its Euler characteristic, we get the divergent series
1 - 1 + 1 - 1 + 1 - ...
But, if we use Leinster's formula, we get 1/2. And that's cute, because
once there were heated arguments about the value of
1 - 1 + 1 - 1 + 1 - ...
Some mathematicians said it was 0:
(1 - 1) + (1 - 1) + (1 - 1) + ... = 0
while others said it was 1:
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = 1
Some said "it's divergent, so forget it!" But others wisely compromised
and said it equals 1/2. This can be justified using "Abel summation".
All this may seem weird - and it is; that's part of the fun. But,
Leinster's answer matches what you'd expect from the theory of "homotopy
cardinality":
6) John Baez, The mysteries of counting: Euler characteristic versus
homotopy cardinality, http://math.ucr.edu/home/baez/counting/
This webpage has transparencies of a talk I gave on this, and lots of
links to papers that generalize the concepts of cardinality and Euler
characteristic. I'm obsessed with this topic. It's really exciting
to think about new ways to extend the simplest concepts of math, like
counting.
That's why I invented a way to compute the cardinality of a groupoid -
a category where every morphism has an inverse, so all the morphisms
describe "symmetries". The idea is that the more symmetries an object
has, the smaller it is. Applying this to the above example, where our
category has one object, and this object has 2 symmetries, one gets 1/2.
If this seems strange, try the explanation in "week147".
Later James Dolan took this idea, generalized it to a large class
of spaces that don't necessarily come from groupoids, and called the
result "homotopy cardinality". We wrote a paper about this.
What Leinster has done is generalize the idea in another direction: from
groupoids to categories. The cool thing is that his generalization
matches the Euler characteristic of spaces coming from categories
(when that's well-defined, without divergent series) and the homotopy
cardinality of spaces coming from groupoids (when that's well-defined).
Of course he doesn't call his thing the "size" of a category; he calls
it the "Euler characteristic" of a category.
Our conversation over coffee was mainly about me trying to understand
the formula he used to define this Euler characteristic. One thing I
learned is that the "category algebra" idea plays a key role here.
It's a simple idea. Given a category X, the category algebra C[X]
consists of all formal complex linear combinations of morphisms in X.
To define the multiplication in this algebra, it's enough to define
the product fg whenever f and g are morphisms in our category. If
the composite of f and g is defined, we just let fg be this composite.
If it's not, we set fg = 0.
Mathematicians seem to be most familiar with the category algebra idea
when our category happens to be a group (a category with one object, all
of whose morphisms are invertible). Then it's called a "group
algebra".
Category algebras are also pretty familiar when our category is a
"quiver" (a category formed from a directed graph by freely throwing
in formal composites of edges). Then it's called a "quiver algebra".
These are really cool - especially if our graph becomes a Dynkin
diagram, like this:
o
|
o--o--o--o--o--o--o
when we ignore the directions of the edges. To see what I mean, try
item E in "week230", where I sketch how these quiver algebras are
related to quantum groups. There's a lot more to say about this, but
not today!
In combinatorics, category algebras are familiar when our category is
a "partially ordered set", or "poset" for short (a category with at
most one morphism from any given object to any other). These category
algebras are usually called "incidence algebras".
In physics, Alain Connes has given a nice explanation of how Heisenberg
invented "matrix mechanics" when he was trying to understand how atoms
jump from one state to another, emitting and absorbing radiation.
In modern language, Heisenberg took a groupoid with n objects, each
one isomorphic to each other in a unique way. He called the objects
"states" of a quantum system, and he called the morphisms "transitions".
Then, he formed its category algebra. The result is the algebra of
n x n matrices!
(This might seem like a roundabout way to get to n x n matrices, but
Heisenberg *didn't know about matrices* at this time. They weren't
part of the math curriculum for physicists back then!)
Connes has generalized the heck out of Heisenberg's idea, studying
the "groupoid algebras" of various groupoids.
So, category algebras are all over the place. But for some reason,
few people study all these different kinds of category algebra in a
unified way - or even *realize* they're all category algebras! I
feel sort of sorry for this neglected concept. That's one reason I
was happy to see it plays a role in Leinster's definition of the Euler
characteristic for categories.
Suppose our category X is finite. Then, we can define an element of
the category algebra C[X] which is just the sum of all the morphisms
in X. This is called zeta, or the "zeta function" of our category.
Sometimes zeta has an inverse, and then this inverse is called mu, or
the "Moebius function" of our category.
Actually, these terms are widely used only when our category is a
poset, thanks to the work of Gian-Carlo Rota, who used these ideas
in combinatorics:
7) Gian-Carlo Rota, On the foundations of combinatorial theory I:
Theory of Moebius Functions, Zeitschrift fuer Wahrscheinlichkeitstheorie
und Verwandte Gebiete 2 (1964), 340-368.
If you want to know what these ideas are good for, check this out:
8) Wikipedia, Incidence algebra,
http://en.wikipedia.org/wiki/Incidence_algebra
See the stuff about Euler characteristics in this article? That's a
clue! The relation to the Riemann zeta function and its inverse
(the original "Moebius function") are clearer here:
9) Wikipedia, Moebius inversion formula,
http://en.wikipedia.org/wiki/Möbius_inversion_formula
These show up when we think of the whole numbers 1,2,3,... as a poset
ordered by divisibility.
Anyway, Leinster has wisely generalized this terminology to more general
categories. And when zeta^{-1} = mu exists, it's really easy to
define his Euler characteristic of the category X. You just write
mu as a linear combination of morphisms in your category, and sum
all the coefficients in this linear combination!
Unfortunately, there are lots of important categories whose zeta
function is not invertible: for example, any group other than the
trivial group. So, Leinster needs a somewhat more general definition to
handle these cases. I don't feel I deeply understand it, but I'll
explain it, just for the record.
Besides the category algebra C[X], consisting of linear combinations
of morphisms in X, there's also a vector space consisting of linear
combinations of *objects* in X. Heisenberg would probably call this
"the space of states", and call C[X] the "algebra of observables",
since that's what they were in his applications to quantum physics.
Let's do that.
The algebra of observables has an obvious left action on the vector
space of states, where a morphism f: x -> y acts on x to give y, and
it acts on every other object to give 0. In Heisenberg's example,
this is precisely how he let the algebra of observables act on states.
The algebra of observables also has an obvious *right* action on the
vector space of states, where f: x -> y acts on y to give x, and it
acts on every other object to give 0.
Leinster defines a "weighting" on X to be an element w of the vector
space of states with
zeta w = 1
Here "1" is the linear combination of objects where all the coefficients
equal 1. He also defines a "coweighting" to be an element w* in the
vector space of states with
w* zeta = 1
If zeta has an inverse, our category has both a weighting and a
coweighting, since we can solve both these equations to find w and w*.
But often there will be a weighting and coweighting even when zeta
doesn't have an inverse. When both a weighting and coweighting exist,
the sum of the coefficients of w equals the sum of coefficients of w* -
and this sum is what Leinster takes as the "Euler characteristic" of the
category X!
This is a bit subtle, and I don't deeply understand it. But, Leinster
proves so many nice theorems about this "Euler characteristic" that
it's clearly the right notion of the size of a category - or, with a
further generalization he mentions, even an n-category! And, it has
nice relationships to other ideas, which are begging to be developed
further.
We're still just learning to count.
-----------------------------------------------------------------------
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
http://math.ucr.edu/home/baez/this.week.html
February 2, 2006
This Week's Finds in Mathematical Physics (Week 244)
John Baez
In January I spent a week at this workshop at the Fields Institute
in Toronto:
1) Higher Categories and Their Applications,
http://math.ucr.edu/home/baez/fields/
It was really fun - lots of people working on n-categories were
there. I'll talk about it next time. But as usual, more happens at
a fun conference than can possibly be reported. So, this time I'll
only talk about a conversation I had in a cafe before the conference
started!
But first, here's a fun way to challenge your math pals:
Q: When the first calculus textbook was written - and in what
language?
A: In 1530, in Malayalam - a south Indian language!
This book is called the Ganita Yuktibhasa, or "compendium of
astronomical rationales". It was written by Jyesthadeva, an
astronomer and mathematician from Kerala - a state on the southwest
coast of India. It summarizes and explains the work of many
researchers of the Kerala school, which flourished from the
1400's to the 1600's. But it's unique for its time, since it
contains proofs of many results.
For example, it has a proof that
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
Of course, this result isn't stated in modern notation! It's
actually stated as a poem - a recipe for the circumference of
a circle, which in translation goes something like this:
"Multiply the diameter by four. Subtract from it and add to it
alternately the quotients obtained by dividing four times the
diameter by the odd numbers 3, 5, etc."
The proof sounds nice! Jyesthadeva starts with something like this:
pi/4 = lim_{N -> infinity} (1/N) Sum_{n=1}^N 1/(1 + (n/N)^2)
In modern terms, the right-hand side is just the integral
integral_0^1 dx/(1 + x^2)
You can use geometry to see this equals pi/4. Then, as far as
I can tell, he writes
1/(1 + (n/N)^2) = 1 - (n/N)^2 + (n/N)^4 - ...
and notes that
1^k + 2^k + ... + N^k ~ N^{k+1}/(k+1)
for large N. This gives
pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...
Voila!
In fact, this result goes back to Madhava, an amazing mathematician
from Kerala who lived much earlier, from 1350 to 1425. What's even
more impressive is that Madhava also knew a formula equivalent to the
more general result
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
He used this to compute pi to 11 decimal places!
It's an interesting question whether any of the results of the
Kerala school found their way west and influenced the development
of mathematics in Europe. There's been a lot of speculation, but
nobody seems to know for sure. For more info, try these:
2) The MacTutor History of Mathematics Archive, Madhava of
Sangamagramma,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Madhava.html
3) The MacTutor History of Mathematics Archive, Jyesthadeva,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Jyesthadeva.html
4) Wikipedia, Yuktibhasa,
http://en.wikipedia.org/wiki/Yuktibhasa
Before the conference started, I spent a nice morning talking with Tom
Leinster in a cafe on Bloor Street. There's nothing like talking
about math in a nice warm cafe when it's cold outside! At some point
my former grad student Toby Bartels showed up - he'd just taken a long
Greyhound bus from Nebraska - and joined in the conversation. We
talked about this paper:
5) Tom Leinster, The Euler characteristic of a category,
available as math.CT/0610260.
Everyone know how to measure the size of a set - by its number
of elements, or "cardinality". But what's the size of a category?
That's the question this paper tackles!
Some categories are just sets in disguise: the "discrete" categories,
whose only morphisms are identity morphisms. We'd better define the
size of such a category to be the cardinality of its set of objects.
For example, the category with just one object and its identity
morphism is called 1. It looks sort of like this:
o
where I've drawn the object but not its identity morphism. Clearly,
its size should be 1.
We could also have a category with just two objects and their identity
morphisms. It looks like this:
o o
and its size should be 2.
But what about this?
o--<--->--o
Here we have a category with two objects and an invertible morphism
between them, which I've drawn as an arrow pointing both ways. Again,
I won't draw the identity morphisms.
In other words, we have two objects that are *isomorphic* - and in a
unique way. How big should this category be?
Any mathematician worth her salt knows that having two things that are
isomorphic in a unique way is just like having one: you can't do
anything more with them - or less. So, the size of this category:
o--<--->--o
should equal the size of this one:
o
namely, 1.
More technically, we say these categories are "equivalent".
We'll demand that equivalent categories have the same size.
This is a powerful principle. If we didn't insist on this,
we'd be insane.
But what about this category:
o---->----o
Now we have two objects and a morphism going just one way!
This is *not* equivalent to a discrete category, so we need
a new idea to define its size.
If we were willing to make up new kinds of numbers, we could
make up a new number for the size of this category. But let's
suppose that this is against the rules.
There's a cute way to turn any category into a space, which I
described in "week70" - and in more detail in items J and K of
"week117", back when I was giving a minicourse on homotopy theory.
If we do this to the category
o---->----o
what do we get? The unit interval, of course! It's a pretty
intuitive notion, at least in this example.
We also get the unit interval if we turn this guy
o--<--->--o
into a space. So, even though these categories aren't equivalent,
they give the same space. So, let's declare that they have the
same size - namely, 1.
In fact, let's adopt this as a new principle! We'll demand that
two categories have the same size whenever they give the same space.
Whenever categories are equivalent, they give the same space (where "the
same" means "homotopy equivalent"). So, our new principle includes our
previous principle as a special case. But, we can say more. If you like
adjoint functors, you'll enjoy this: whenever there's a pair of adjoint
functors going between two categories, they give the same space. For
example, these categories
o---->----o
and
o--<--->--o
aren't equivalent, but there's a pair of adjoint functors going between
them. (If you don't like adjoint functors, oh well - just ignore this.)
Next, what's the size of this category?
---->----
/ \
o o
\ /
---->----
This is my feeble attempt to draw a category with two objects, and two
morphisms going from the first object to the second.
If we turn this category into the space, what do we get? The circle, of
course! But what's the "size", or "cardinality", of a circle?
That's a tricky puzzle, because it's hard to know what counts as a right
answer. It turns out the right answer is zero. Why? Because the
"Euler characteristic" of the circle is zero!
As you may know, Euler lived in Konigsberg, a city with lots of islands
and bridges. In fact, he published a paper in 1736 showing that you
can't walk around Konigsberg and cross each bridge exactly once, winding
up where you started. My crazy theory is that living there also helped
him invent the concept of Euler characteristic. I have no evidence for
this, except for this apocryphal story I just made up:
Once upon a time, Euler was strolling along one of the bridges of
Konigsberg. He looked across the river, and noticed that workers
werebuilding a bridge to a small island that had previously been
unconnected to the rest. He noticed that this reduced the number of
isolated islands by one. Of course, anyone could have seen that!
But in a burst of genius, Euler went further - he realized this meant a
bridge was like a "negative island". And so, he invented the concept
of "Euler characteristic". In its simplest form, it's just the number
of islands minus the number of bridges.
For example, if you have two islands in the sea:
o o
the land has Euler characteristic 2.
If you build a bridge:
o--<--->--o
the land now has Euler characteristic 1. This makes sense, because the
land is now effectively just one island. So, a bridge acts as a
"negative island"!
But now, if you build a *second* bridge:
---->----
/ \
o o
\ /
---->----
the land has Euler characteristic 0. This is sort of weird. But, Euler
saw it was a good idea.
To understand why, you have to go further and imagine building a "bridge
between bridges" - filling in the space between the bridges with an
enormous deck:
---->----
/xxxxxxxxx\
oxxxxxxxxxxxo
\xxxxxxxxx/
---->----
This reduces the number of bridges by one. We've effectively got one
island again, though much bigger now. So, we're back to having Euler
characteristic 1.
In short, adding a "bridge between bridges" should add 1 to the Euler
characteristic. Just as a bridge counts as a negative island, a bridge
between bridges counts as a negative bridge - or an island:
-(-1) = 1.
It's all consistent, in its own weird way.
So, Euler defined the Euler characteristic to be
V - E + F
where V is the number of islands (or "vertices"), E is the number of
bridges (or "edges") and F is the number of bridges between bridges (or
"faces").
At least that's how the story goes.
By the way, you must have noticed that the number 1 looks like an
interval, while the number 0 looks like a circle. But did you notice
that the Euler characteristic of the interval is 1, and the Euler
characteristic of the circle is 0? I can never make up my mind whether
this is a coincidence or not.
Anyway, we can easily generalize the Euler characteristic to higher
dimensions, and define it as an alternating sum. And that turns out
to be important for us now, because it turns out that often when we
turn a category into a space, we get something higher-dimensional!
This shouldn't be obvious, since I haven't told you the rule for turning
a category into a space. You might think we always get something
1-dimensional, built from vertices (objects) and edges (morphisms).
But the rule is more subtle. Whenever we have 2 morphisms end to end,
like this:
f g
o--->---o--->---o
X Y Z
we can compose them and get a morphism fg going all the way from x to z.
We should draw this morphism too... so the space we get is a *triangle*:
Y
o
/x\
f /xxx\ g
/xxxxx\
X o-------o Z
fg
I haven't drawn the arrows on my morphisms, due to technical limitations
of this medium. More importantly, the triangle is filled with x's, just
like Euler's "bridge between bridges", to show that it's *solid*, not
hollow.
Simlarly, when we have 3 morphisms laid end to end we get a tetrahedron,
and so on.
Using these rules, it's not hard to find a category that gives a sphere,
or a torus, or an n-holed torus, when you turn it into a space. I'll
leave that as a puzzle.
In fact, for *any* manifold, you can find a category that gives you
that manifold when you turn it into a space! In fact we can get any
space at all this way, up to "weak homotopy equivalence" - whatever
that means. So, let's adopt a new principle: whenever our category
gives a space whose Euler characteristic is well-defined, we should
define the size of our category to be that.
I say "when it's well-defined", because it's also possible for a
category - even one with just finitely many objects and morphisms - to
give an infinite-dimensional space whose Euler characteristic is a
divergent series:
n_0 - n_1 + n_2 - n_3 + n_4 - ...
Okay. At this point it's time for me to say what Leinster actually did:
he came up with a *formula* that you can use to compute the size of a
category, without using any topology. Sometimes it gives divergent
answers - which is no shame: after all, some categories are infinitely
big. But when it converges, it satisfies all the principles I've
mentioned.
Even better, it works for a lot of categories that give spaces whose
Euler chacteristic diverges! For example, we can take any group G and
think of it as a category with one object, with the group elements as
morphisms. When we turn this category into a space, it becomes
something famous called the "classifying space" of G. This is often
an infinite-dimensional monstrosity whose Euler characteristic diverges.
But, Leinster's formula still works - and it gives
1/|G|
the reciprocal of the usual cardinality of G.
Now we're getting fractions!
For example, suppose we take G to be the group with just 2 elements,
called Z/2. If we think of it as a category, and then turn that into
a space, we get a huge thing usually called "infinite-dimensional real
projective space", or RP^infinity for short. This is built from one
vertex, one edge, one triangle, and so on. So, if we try to work out
its Euler characteristic, we get the divergent series
1 - 1 + 1 - 1 + 1 - ...
But, if we use Leinster's formula, we get 1/2. And that's cute, because
once there were heated arguments about the value of
1 - 1 + 1 - 1 + 1 - ...
Some mathematicians said it was 0:
(1 - 1) + (1 - 1) + (1 - 1) + ... = 0
while others said it was 1:
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = 1
Some said "it's divergent, so forget it!" But others wisely compromised
and said it equals 1/2. This can be justified using "Abel summation".
All this may seem weird - and it is; that's part of the fun. But,
Leinster's answer matches what you'd expect from the theory of "homotopy
cardinality":
6) John Baez, The mysteries of counting: Euler characteristic versus
homotopy cardinality, http://math.ucr.edu/home/baez/counting/
This webpage has transparencies of a talk I gave on this, and lots of
links to papers that generalize the concepts of cardinality and Euler
characteristic. I'm obsessed with this topic. It's really exciting
to think about new ways to extend the simplest concepts of math, like
counting.
That's why I invented a way to compute the cardinality of a groupoid -
a category where every morphism has an inverse, so all the morphisms
describe "symmetries". The idea is that the more symmetries an object
has, the smaller it is. Applying this to the above example, where our
category has one object, and this object has 2 symmetries, one gets 1/2.
If this seems strange, try the explanation in "week147".
Later James Dolan took this idea, generalized it to a large class
of spaces that don't necessarily come from groupoids, and called the
result "homotopy cardinality". We wrote a paper about this.
What Leinster has done is generalize the idea in another direction: from
groupoids to categories. The cool thing is that his generalization
matches the Euler characteristic of spaces coming from categories
(when that's well-defined, without divergent series) and the homotopy
cardinality of spaces coming from groupoids (when that's well-defined).
Of course he doesn't call his thing the "size" of a category; he calls
it the "Euler characteristic" of a category.
Our conversation over coffee was mainly about me trying to understand
the formula he used to define this Euler characteristic. One thing I
learned is that the "category algebra" idea plays a key role here.
It's a simple idea. Given a category X, the category algebra C[X]
consists of all formal complex linear combinations of morphisms in X.
To define the multiplication in this algebra, it's enough to define
the product fg whenever f and g are morphisms in our category. If
the composite of f and g is defined, we just let fg be this composite.
If it's not, we set fg = 0.
Mathematicians seem to be most familiar with the category algebra idea
when our category happens to be a group (a category with one object, all
of whose morphisms are invertible). Then it's called a "group
algebra".
Category algebras are also pretty familiar when our category is a
"quiver" (a category formed from a directed graph by freely throwing
in formal composites of edges). Then it's called a "quiver algebra".
These are really cool - especially if our graph becomes a Dynkin
diagram, like this:
o
|
o--o--o--o--o--o--o
when we ignore the directions of the edges. To see what I mean, try
item E in "week230", where I sketch how these quiver algebras are
related to quantum groups. There's a lot more to say about this, but
not today!
In combinatorics, category algebras are familiar when our category is
a "partially ordered set", or "poset" for short (a category with at
most one morphism from any given object to any other). These category
algebras are usually called "incidence algebras".
In physics, Alain Connes has given a nice explanation of how Heisenberg
invented "matrix mechanics" when he was trying to understand how atoms
jump from one state to another, emitting and absorbing radiation.
In modern language, Heisenberg took a groupoid with n objects, each
one isomorphic to each other in a unique way. He called the objects
"states" of a quantum system, and he called the morphisms "transitions".
Then, he formed its category algebra. The result is the algebra of
n x n matrices!
(This might seem like a roundabout way to get to n x n matrices, but
Heisenberg *didn't know about matrices* at this time. They weren't
part of the math curriculum for physicists back then!)
Connes has generalized the heck out of Heisenberg's idea, studying
the "groupoid algebras" of various groupoids.
So, category algebras are all over the place. But for some reason,
few people study all these different kinds of category algebra in a
unified way - or even *realize* they're all category algebras! I
feel sort of sorry for this neglected concept. That's one reason I
was happy to see it plays a role in Leinster's definition of the Euler
characteristic for categories.
Suppose our category X is finite. Then, we can define an element of
the category algebra C[X] which is just the sum of all the morphisms
in X. This is called zeta, or the "zeta function" of our category.
Sometimes zeta has an inverse, and then this inverse is called mu, or
the "Moebius function" of our category.
Actually, these terms are widely used only when our category is a
poset, thanks to the work of Gian-Carlo Rota, who used these ideas
in combinatorics:
7) Gian-Carlo Rota, On the foundations of combinatorial theory I:
Theory of Moebius Functions, Zeitschrift fuer Wahrscheinlichkeitstheorie
und Verwandte Gebiete 2 (1964), 340-368.
If you want to know what these ideas are good for, check this out:
8) Wikipedia, Incidence algebra,
http://en.wikipedia.org/wiki/Incidence_algebra
See the stuff about Euler characteristics in this article? That's a
clue! The relation to the Riemann zeta function and its inverse
(the original "Moebius function") are clearer here:
9) Wikipedia, Moebius inversion formula,
http://en.wikipedia.org/wiki/Möbius_inversion_formula
These show up when we think of the whole numbers 1,2,3,... as a poset
ordered by divisibility.
Anyway, Leinster has wisely generalized this terminology to more general
categories. And when zeta^{-1} = mu exists, it's really easy to
define his Euler characteristic of the category X. You just write
mu as a linear combination of morphisms in your category, and sum
all the coefficients in this linear combination!
Unfortunately, there are lots of important categories whose zeta
function is not invertible: for example, any group other than the
trivial group. So, Leinster needs a somewhat more general definition to
handle these cases. I don't feel I deeply understand it, but I'll
explain it, just for the record.
Besides the category algebra C[X], consisting of linear combinations
of morphisms in X, there's also a vector space consisting of linear
combinations of *objects* in X. Heisenberg would probably call this
"the space of states", and call C[X] the "algebra of observables",
since that's what they were in his applications to quantum physics.
Let's do that.
The algebra of observables has an obvious left action on the vector
space of states, where a morphism f: x -> y acts on x to give y, and
it acts on every other object to give 0. In Heisenberg's example,
this is precisely how he let the algebra of observables act on states.
The algebra of observables also has an obvious *right* action on the
vector space of states, where f: x -> y acts on y to give x, and it
acts on every other object to give 0.
Leinster defines a "weighting" on X to be an element w of the vector
space of states with
zeta w = 1
Here "1" is the linear combination of objects where all the coefficients
equal 1. He also defines a "coweighting" to be an element w* in the
vector space of states with
w* zeta = 1
If zeta has an inverse, our category has both a weighting and a
coweighting, since we can solve both these equations to find w and w*.
But often there will be a weighting and coweighting even when zeta
doesn't have an inverse. When both a weighting and coweighting exist,
the sum of the coefficients of w equals the sum of coefficients of w* -
and this sum is what Leinster takes as the "Euler characteristic" of the
category X!
This is a bit subtle, and I don't deeply understand it. But, Leinster
proves so many nice theorems about this "Euler characteristic" that
it's clearly the right notion of the size of a category - or, with a
further generalization he mentions, even an n-category! And, it has
nice relationships to other ideas, which are begging to be developed
further.
We're still just learning to count.
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