Date: Jan 6, 2003 2:25 AM
Author: John Baez
Subject: Re: Queries about Species

In article <av8rch$bki$>,
David Corfield <> wrote:

>I have a few questions about combinatorics and species, and would be
>grateful for any comments.

I'll tackle the easiest one now and attempt the harder ones
later, but I sure hope other people try too.

>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.

I don't see what you're worrying about here, but I presume
it's related to the naive "0/0" you get when you evaluate
this expression at X= 0.

I've been meaning to think about this ever since I read Connes'
comments on Bernoulli numbers in this book:

Alain Connes, Andre Lichnerowicz and Marcel Paul Schutzenberger,
A Triangle of Thoughts, AMS, Providence, 2000.

He points out that if H is the Hamiltonian for some sort
of particle in a box and beta is the inverse temperature,

1/(1 - e^{-beta H}) = 1 + e^{-beta H} + e^{-2 beta H} + ...

is the operator you take the trace of to get the partition
function of a collection of an arbitrary number of particles of
this sort. And he claims that pondering this explains all the
appearances of X/(1 - e^X) and the Bernoulli numbers in topology!
See Milnor and Stasheff's book "Characteristic Classes" for an
introduction to *that* - but this book was written before
quantum theory invaded topology, so we're left to fit Connes'
clues together for ourselves.

>Why is it that you get the series expansions for species if they don't
>blow up for X= 0, yet you're most interested in X= 1?

Well, for now I'll just say that that species don't "blow up";
it's only when you decategorify them that you get divergent formal
power series. Then of course they usually diverge at X = 1,
because we are usually interested in structures that can be
put on finite sets in infinitely many different ways.

You probably knew all this and wanted a deeper answer;
I don't think there is one. I forget if you know what it
means to evaluate a species at an arbitrary groupoid X;
if not, maybe this would make you happier.