From: baez@math.ucr.edu (John Baez) Subject: This Week's Finds in Mathematical Physics (Week 147) Date: Sat, 20 May 2000 11:30:01 -0500 (CDT) Newsgroups: sci.math.research Summary: [missing] Keywords: Also available at http://math.ucr.edu/home/baez/week147.html This Week's Finds in Mathematical Physics (Week 147) John Baez Various books are coming out to commemorate the millennium.... describing the highlights of the math we've done so far, and laying out grand dreams for the future. The American Mathematical Society has come out with one: 1) Mathematics: Frontiers and Perspectives, edited by Vladimir Arnold, Michael Atiyah, Peter Lax and Barry Mazur, AMS, Providence, Rhode Island, 2000. This contains 30 articles by bigshots like Chern, Connes, Donaldson, Jones, Lions, Manin, Mumford, Penrose, Smale, Vafa, Wiles and Witten. I haven't actually read it yet, but I want to get ahold of it. Springer Verlag is coming out with one, too: 2) Mathematics Unlimited: 2001 and Beyond, edited by Bjorn Engquist and Wilfried Schmid, Springer Verlag, New York, 2000. It should appear in the fall. I don't know what the physicists are doing along these lines. The American Physical Society has a nice timeline of 20th century physics on their website: 3) The American Physical Society: A Century of Physics, available at http://timeline.aps.org/APS/home_HighRes.html But I don't see anything about books. One reason I haven't been doing many This Week's Finds lately is that I've been buying and then moving into a new house. Another is that James Dolan and I have been busily writing our own millennial pontifications, which will appear in the Springer-Verlag book: 4) John Baez and James Dolan, From finite sets to Feynman diagrams, preprint available as math.QA/0004133. So let me talk about this stuff a bit.... As usual, the underlying theme of this paper is categorification. I've talked about this a lot already - e.g. in "week121" - so I'll assume you remember that when we categorify, we use this analogy: SET THEORY CATEGORY THEORY elements objects equations between elements isomorphisms between objects sets categories functions functors equations between functions natural isomorphisms between functors to take interesting *equations* and see them as shorthand for even more interesting *isomorphisms*. To take a simple example, consider the laws of basic arithmetic, like a+b = b+a or a(b+c) = ab+ac. We usually think of these as *equations* between *elements* of the *set* of natural numbers. But really they arise from *isomorphisms* between *objects* of the *category* of finite sets. For example, if we have finite sets a and b, and we use a+b to denote their disjoint union, then there is a natural isomorphism between a+b and b+a. Moreover, this isomorphism is even sort of interesting! For example, suppose we use 1 to denote a set consisting of one dot, and 2 to denote a set of two dots. Then the natural isomorphism between 1+2 and 2+1 can be visualized as the process of passing one dot past two, like this: . . . \ / / \ / / \ / / / / / \ / / / / / \ / / \ / / \ / / \ . . . This may seem like an excessively detailed "picture proof" that 1+2 indeed equals 2+1, perhaps suitable for not-too-bright kindergarteners. But in fact it's just a hop, skip and a jump from here to heavy-duty stuff like the homotopy groups of spheres. I sketched how this works in "week102" so I won't do so again here. The point is, after we categorify, elementary math turns out to be pretty powerful! Now, let me make this idea of "categorifying the natural numbers" a bit more precise. Let FinSet stand for the category whose objects are finite sets and whose morphisms are functions between these. If we "decategorify" this category by forming the set of isomorphism classes of objects, we get N, the natural numbers. All the basic arithmetic operations on N come from operations on FinSet. I've already noted how addition comes from disjoint union. Disjoint union is a special case of what category theorists call the "coproduct", which makes sense for a bunch of categories - see "week99" for the general definition. Similarly, multiplication comes from the Cartesian product of finite sets, which is a special case of what category theorists call the "product". To get the definition of a product, you just take the definition of a coproduct and turn all the arrows around. There are also nice category-theoretic interpretations of the numbers 0 and 1, and all the basic laws 0, 1, addition and multiplication. Exponentiation too! Combinatorists have lots of fun thinking about how to take equations in N and prove them using explicite isomorphisms in FinSet - they call such a proof a "bijective proof". To read more about this, try: 4) James Propp and David Feldman, Producing new bijections from old, Adv. Math. 113 (1995), 1-44. Also available at http://www.math.wisc.edu/~propp/articles.html 5) John Conway and Peter Doyle, Division by three. http://math.dartmouth.edu/~doyle/docs/three/three/three.html The latter article studies this question: if I give you an isomorphism between 3x and 3y, can you construct a isomorphism between x and y? Here of course x and y are finite sets, 3 is any 3-element set, and multiplication means Cartesian product. Of course you can prove an isomorphism exists, but can you *construct* one in a natural way - i.e., without making any random choices? The history of this puzzle turns out to be very interesting. But I don't want to give away the answer! See if you can do it or not. Anyway, having categorified the natural numbers, we might be inclined to go on and categorify the integers. Can we do it? In other words: can we find something like the category of finite sets that includes "sets with a negative number of elements"? There turns out be an interesting literature on this subject: 6) Daniel Loeb, Sets with a negative number of elements, Adv. Math. 91 (1992), 64-74. 7) S. Schanuel, Negative sets have Euler characteristic and dimension, Lecture Notes in Mathematics 1488, Springer Verlag, Berlin, 1991, pp. 379-385. 8) James Propp, Exponentiation and Euler characteristic, http://www.math.wisc.edu/~propp/articles.html 9) Andre Joyal, Regle des signes en algebre combinatoire, Comptes Rendus Mathmatiques de l'Academie des Sciences, La societe royale du Canada, VII (1985), 285-290. See also "week102" for more.... But I don't want to talk about negative sets right now! Instead, I want to talk about *fractional* sets. It may seem odd to tackle division before subtraction, but historically, the negative numbers were invented quite a bit *after* the nonnegative rational numbers. Apparently half an apple is easier to understand than a negative apple! This suggests that perhaps `sets with fractional cardinality' are simpler than `sets with negative cardinality'. The key is to think carefully about the meaning of division. The usual way to get half an apple is to chop one into `two equal parts'. Of course, the parts are actually NOT EQUAL --- if they were, there would be only one part! They are merely ISOMORPHIC. This suggests that categorification will be handy. Indeed, what we really have is a Z/2 symmetry group acting on the apple which interchanges the two isomorphic parts. In general, if a group G acts on a set S, we can `divide' the set by the group by taking the quotient S/G, whose points are the orbits of the action. If S and G are finite and G acts freely on S, this construction really corresponds to division, since the cardinality |S/G| is equal to |S|/|G|. However, it is crucial that the action be free. For example, why is 6/2 = 3? We can take a set S consisting of six dots in a row: o o o o o o let G = Z/2 act freely by reflections, and identify all the elements in each orbit to obtain 3-element set S/G. Pictorially, this amounts to folding the set S in half, so it is not surprising that |S/G| = |S|/|G| in this case. Unfortunately, if we try a similar trick starting with a 5-element set: o o o o o it fails miserably! We don't obtain a set with 2.5 elements, because the group action is not free: the point in the middle gets mapped to itself. So to define `sets with fractional cardinality', we need a way to count the point in the middle as `half a point'. To do this, we should first find a better way to define the quotient of S by G when the action fails to be free. Following the policy of replacing equations by isomorphisms, let us define the "weak quotient" S//G to be the category with elements of S as its objects, with a morphism g: s -> s' whenever g(s) = s', and with composition of morphisms defined in the obvious way. Next, let us figure out a good way to define the `cardinality' of a category. Pondering the examples above leads us to the following recipe: for each isomorphism class of objects we pick a representative x and compute the reciprocal of the number of automorphisms of this object; then we sum over isomorphism classes. It is easy to see that with this definition, the point in the middle of the previous picture gets counted as `half a point' because it has two automorphisms, so we get a category with cardinality 2.5. In general, S//G = |S|/|G| whenever G is a finite group acting on a finite set S. This formula is a simplified version of `Burnside's lemma', so-called because it is due to Cauchy and Frobenius. Burnside's lemma gives the cardinality of the ordinary quotient. But the weak quotient is nicer, so Burnside's lemma simplifies when we use weak quotients. Now, the formula for the cardinality of a category makes sense even for some categories that have infinitely many objects - all we need is for the sum to make sense. So let's try to compute the cardinality of the category of finite sets! Since any n-element set has n! automorphisms (i.e. permutations), we get following marvelous formula: |FinSet| = e This turns out to explain lots of things about the number e. Now, a category all of whose morphisms are isomorphisms is called a "groupoid". Any category C has an underlying groupoid C_0 with the same objects but only the isomorphisms as morphisms. The cardinality of a category C always equals that of its underlying groupoid C_0. This suggests that this notion should really be called "groupoid cardinality. If you're a fan of n-categories, this suggests that we should generalize the concept of cardinality to n-groupoids, or even omega-groupoids. And luckily, we don't need to understand omega-groupoids very well to try our hand at this! Omega-groupoids are supposed to be an algebraic way of thinking about topological spaces up to homotopy. Thus we just need to invent a concept of the `cardinality' of a topological space which has nice formal properties and which agrees with the groupoid cardinality in the case of homotopy 1-types. In fact, this is not hard to do. We just need to use the homotopy groups pi_k(X) of the space X. So: let's define the "homotopy cardinality" of a topological space X to be the alternating product |X| = |pi_1(X)|^{-1} |pi_2(X)| |pi_3(X)|^{-1} .... when X is connected and the product converges; if X is not connected, let's define its homotopy cardinality to be the sum of the homotopy cardinalities of its connected components, when the sum converges. We call spaces with well-defined homotopy cardinality "tame". The disjoint union or Cartesian product of tame spaces is again tame, and we have |X + Y| = |X| + |Y| , |X x Y| = |X| x |Y| just as you would hope. Even better, homotopy cardinality gets along well with fibrations, which we can think of as `twisted products' of spaces. Namely, if F -> X -> B is a fibration and the base space B is connected, we have |X| = |F| |B| whenever two of the three spaces in question are tame (which implies the tameness of the third). As a fun application of this fact, recall that any topological group G has a "classifying space" BG, meaning a space with a principal G-bundle over it G -> EG -> BG whose total space EG is contractible. I described how to construct the classifying space in "week117", at least in the case of a discrete group G, but I didn't say much about why it's so great. The main reason it's great is that *any* G-bundle over *any* space is a pullback of the bundle EG over BG. But right now, what I want to note is that since EG is contractible it is tame, and |EG| = 1. Thus G is tame if and only if BG is, and |BG| = 1 / |G| so we can think of BG as the `reciprocal' of G! This idea is already lurking behind the usual approach to "equivariant cohomology". Suppose X is a space on which the topological group G acts. When the action of G on X is free, it is fun to calculate cohomology groups (and other invariants) of the quotient space X/G. When the action is not free, this quotient can be very pathological, so people usually replace it by the "homotopy quotient" X//G, which is defined as (EG x X)/G. This is like the ordinary quotient but with equations replaced by homotopies. And there is a fibration X -> X//G -> BG , so when X and G are tame we have |X//G| = |X| x |BG| = |X|/|G| just as you would hope! Now in the paper, Jim and I go on to talk about how all these ideas can be put to use to give a nice explanation of the combinatorics of Feynman diagrams. But I don't want to explain all that stuff here - then you wouldn't need to read the paper! Instead, I just want to point out something mysterious about homotopy cardinality. Homotopy cardinality is formally very similar to Euler characteristic. The Euler characteristic c(X) is given by the alternating sum c(X) = dim(H_0(X)) - dim(H_1(X)) + dim(H_2(X)) - .... whenever the sum converges, where H_n(X) is a vector space over the rational numbers called the nth rational homology group of X. Just as for homotopy cardinality, we have c(X + Y) = c(X) + c(Y), c(X x Y) = c(X) x c(Y) and more generally, whenever F -> X -> B is a fibration and the base space B is connected, we have c(X) = c(F) c(B) whenever any two of the spaces have well-defined Euler characteristic, which implies that the third does too (unless I'm confused). So Euler characteristic is a lot like homotopy cardinality. But not many spaces have *both* well-defined homotopy cardinality *and* well-defined Euler characteristic. So they're like Jekyll and Hyde - you hardly ever see them in the same place at the same time, so you can't tell if they're really the same guy. But there are some weird ways to try to force the issue and compute both quantities for certain spaces. For example, suppose G is a finite group. Then we can build BG starting from a simplicial set with 1 nondegenerate 0-simplex, |G|-1 nondegenerated 1-simplices, (|G|-1)^2 nondegenerate 2-simplices, and so on. If there were only finitely many simplices of all dimensions, we could compute the Euler characteristic of this space as the alternating sum of the numbers of simplices in each dimension. So let's try doing that here! We get: c(BG) = 1 + (|G|-1) + (|G|-1)^2 + .... Of course the sum diverges, but if we go ahead and use the geometric formula anyway, we get c(BG) = 1/|G| which matches our previous (rigorous) result that |BG| = 1/|G| So maybe they're the same after all! There are similar calculations like this in James Propp's paper "Exponentiation and Euler characteristic", referred to above... though he uses a slightly different notion of Euler characteristic, due to Schanuel. Clearly something interesting is going on with these "divergent Euler characteristics". For appearances of this sort of thing in physics, see: 10) Matthias Blau and George Thompson, N = 2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant, Comm. Math. Phys. 152 (1993), 41-71. and the references therein. (I discussed this paper a bit in "week51".) However, there are still challenging tests to the theory that homotopy cardinality and Euler characteristic are secretly the same. Here's a puzzle due to James Dolan. Consider a Riemann surface of genus g > 1. Such a surface has Euler characteristic 2 - 2g, but such a surface also has vanishing homotopy groups above the first, which implies that it's BG for G equal to its fundamental group. If homotopy cardinality and Euler characteristic were the same, this would imply |G| = 1/|BG| = 1/c(G) = 1/(2 - 2g) But the fundamental group G is infinite! What's going on? Well, I'm actually sort of glad that 1/(2 - 2g) is *negative*. Sometimes a divergent series of positive integers can be cleverly summed up to give a negative number. The simplest example is the geometric series 1 + 2 + 4 + 8 + 16 + ... = 1/(1 - 2) = -1 but in "week126" I talked about a more sophisticated example that is very important in string theory: 1 + 2 + 3 + 4 + 5 + ... = zeta(-1) = -1/12 So maybe some similar trickery can be used to count the elements of G and get a divergent sum that can be sneakily evaluated to obtain 1/(2 - 2g). Of course, even if we succeed in doing this, the skeptics will rightly question the significance of such tomfoolery. But there is sometimes a lot of profound truth lurking in these bizarre formal manipulations, and sometimes if you examine what's going on carefully enough, you can discover cool stuff. To wrap up, let me mention an interesting paper on the foundations of categorification: 11) Claudio Hermida, From coherent structures to universal properties, available at http://www.cs.math.ist.utl.pt/cs/s84/claudio.html and also two papers about 2-groupoids and topology: 12) K.A. Hardie, K.H. Kamps, R.W. Kieboom, A homotopy bigroupoid of a topological space, in: Categorical Methods in Algebra and Topology, pp. 209-222, Mathematik-Arbeitspapiere 48, Universitaet Bremen, 1997. Appl. Categ. Structures, to appear. K.A. Hardie, K.H. Kamps, R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, preprint. I would talk about these if I had the energy, but it's already past my bed-time. Good night! ----------------------------------------------------------------------- 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/twf.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