In article <email@example.com>, David Corfield <firstname.lastname@example.org> wrote:
>I have a few questions about combinatorics and species, and would be >grateful for any comments.
I'll take another crack at these now. By the way, you might have gotten more replies if you'd phrased your question in terms of *generating functions*, rather than *species*. Category theory is great, but most people don't know this, so when trying to extract information from them it's often best to play down this aspect of things, emphasizing other buzzwords.
>X/(1 - e^X) looks like a simple composition of species - pick out a one >element set and arrange a set of sets whose union is the remainder - yet >it can't be that simple to get at the Bernoulli numbers. I guess lots of >unwanted empty sets appear in the union. This would no doubt require >some of Cartier's mathemagics to make sense of it.
Okay, I understand this remark now that you've expanded it... and resummed to get a finite answer.
I'm getting more and more confused and tantalized by the relation between Bernoulli numbers, species, statistical mechanics and quantum theory. The reason I mentioned Connes' remarks is that he was emphasizing the relation between Bernoulli numbers and *partition functions*. Partition functions in statistical mechanics and quantum theory are often just another way of looking at generating functions of species. If you haven't thought about this yet, I can perhaps shed a little light on what's going on - though not enough to fully penetrate all the murk, yet.
If you have a system with allowed energy levels E_n, the probability that it's in the nth one is proportional to
where b, usually written "beta", is the inverse temperature - or really 1/kT where k is Boltzmann's constant and T is the temperature. To get probabilities we need to normalize these numbers by dividing by the "partition function"
Z(b) = sum_n exp(-b E_n)
You can do lots of great stuff in statistical mechanics if you know the partition function of a system, mainly by differentiating it with respect to b and playing various games.
Now, if we don't count the "zero-point energy", the harmonic oscillator in quantum mechanics has allowed energy levels 0,1,2,3,4,... , so its partition function is
Z(b) = sum_n exp(-nb)
1 = ------------- 1 - exp(-b)
This is where Connes' remark about Bernoulli numbers comes in. The above partition function blows up as b -> 0 (the infinite temperature limit, where all energy levels become equally probable). In fact it has a first-order pole at b = 0. To deal with this, we can work with
b b Z(b) = ------------- 1 - exp(-b)
This is analytic at b = 0, and its Taylor series defines the Bernoulli numbers, up to some signs:
b Z(b) = sum B_n (-x)^n / n!
What does all this have to do with species? I'm not sure. But it's tantalizing, for two reasons! In my work with Jim Dolan, the harmonic oscillator itself is categorified: its Hilbert space becomes the category of species, and its Hamiltonian (the "number operator") becomes a functor from this category to itself. In your post, you try to interpret B_n as the cardinality of a set of structures on the n-element set. You get get a divergent series, basically the Riemann zeta function evaluated at a negative integer, and you resum this to get the right answer. This trick is common in quantum field theory under the name of "zeta function regularization".
So, there seem to be two possible connections between Bernoulli numbers and species via physics, but they don't fit together neatly, and neither one is fully worked out - though yours comes a lot closer.
>Now, if a species is a categorification of an element of N[[x]], why are, >Fiore and Leinster interested in categorifying N[x]/ (some polynomial in x), >where the polynomial is a relation for the species?
Yikes... another twisted loop of ideas?
Here's all I can say:
We can indeed define "multivariable species" that serve as a categorification of rigs like N[[x_1,...,x_n]]. To do this, just define an "n-variable species" to be a type of structure that we can put on an ordered n-tuples of finite sets. In other words, it's a functor
F: FinSet_0 x ... FinSet_0 -> Set
where FinSet_0 is the groupoid of finite sets and bijections, and we take an n-fold Cartesian product of this category.
So, yes, we could define "complex species" whose generating functions have complex coefficients by starting with two-variable species and then imposing Fiore and Leinster's equation on one variable to make that variable act like "i".