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This Week's Finds in Mathematical Physics (Week 244)

  1. Feb 14, 2007 #1
    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,

    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

    But first, here's a fun way to challenge your math pals:

    Q: When the first calculus textbook was written - and in what

    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 + ...


    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

    3) The MacTutor History of Mathematics Archive, Jyesthadeva,

    4) Wikipedia, 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:


    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?


    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:


    should equal the size of this one:


    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:


    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


    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


    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




    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:


    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:


    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

    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
    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*:

    f /xxx\ g
    X o-------o Z

    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

    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

    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


    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

    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

    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

    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:


    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,

    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,

    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

    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


    For a table of contents of all the issues of This Week's Finds, try


    A simple jumping-off point to the old issues is available at


    If you just want the latest issue, go to

  2. jcsd
  3. Feb 15, 2007 #2
    baez@math.removethis.ucr.andthis.edu (John Baez) writes:
    > 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?

    How about the same size as the set of complex 4th roots of unity?

    "Home taping is killing big business profits. We left this side blank
    so you can help." -- Dead Kennedys, written upon the B-side of tapes of
    /In God We Trust, Inc./.
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