Followup on my earlier claim to have been categorified.
The Cohen-Lenstra conjectures govern the p-adic variation of class numbers of quadratic imaginary fields. At first glance they look very strange. If you ask the conjectures what proportion of quadratic imaginary class numbers are indivisible by 3, you don’t get 2/3, as you might expect if “class numbers were random numbers,” but rather the infinite product
It gets worse — if you ask for the probability that the class number is indivisible by 27, you get the same infinite product multiplied by some completely meaningless-looking rational function in 1/3.
And you ask: how could anyone ever come up with these conjectures?
The answer, of course, is that they didn’t think about class numbers at all — they thought about class groups. And in that language, their conjecture is clean to the point of being self-explanatory: each finite abelian p-group G appears as the p-primary part of a class group with probability inversely proportional to |Aut(G)|.
Numbers are great; but in this context they are merely the Grothendieck K_0 of the category of finite abelian groups. If life gives you finite abelian groups, use them. To pass to the Grothendieck group is to decategorify. And to decategorify is to tempt fate.
[…] class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that […]