Subject: categories: On Baez on 'complex Euler characteristics'

From: "Stephen Schanuel" <[email-masked]>
Date: Thu, 26 Feb 2004 00:11:23 -0500

Dear colleagues,

    For those who read the most recent long discursion of John Baez, a few of the errors in the section on distributive categories merit correction:

(1) J. B. suggests that Blass published what Lawvere had already worked out. In fact, Lawvere (partly to counteract some incorrect uses of infinite series in  analyses of 'data types' in computer science) had worked out the algebra of the rig presented by one generator X and one relation X=1+X^2, roughly by the method in (3) below, and conjectured that this rig could be realized as the isomorphism classes in a distributive (even extensive) category, which conjecture Blass then proved (and a bit more) in "Seven Trees...".

(2) The generalization of Blass's theorem to one generator ond one polynomial relation of the 'fixed-point' form X=p(X), where p is a polynomial with natural number coefficients and nonzero constant term is not, as J. B. seems to suggest, due to Fiore and Leinster; it was part of the prize-winning doctoral thesis of Robbie Gates, who (using a calculus of fractions) described explicitly the free distributive category on one object X together with an isomorphism from p(X) to X, proving that this category is extensive and that its rig of isomorphism classes satisfies no further relations, i.e. is the rig R presented by one generator and the one relation above.

(3) If p is as in (2) and of degree at least 2, the algebra of the rig R is made by J. B. to seem mysterious. It is more easily understood in the way the X=2X+1 case was treated in my "Negative Sets..." paper; just show that the Euler and dimension homomorphisms, tensoring with Z and with 2 (the rig true/false) respectively, are jointly injective. In this case the dimension rig has only three elements, which explains why the Euler characteristic captures almost, but not quite, everything.

Greetings to all,
    Steve Schanuel