While reading the introduction to this paper by Curtis McMullen, I came to the following (bold added):

In this paper we show that every bifurcation set contains a copy of the boundary of the Mandelbrot set or its degree $d$ generalization. The Mandelbrot sets $M_d$ are thus universal; they are initial objects in the category of bifurcations, providing a lower bound on the complexity of $B(f)$ for all families $f_t$.

(Here $f_t$ is a family of rational functions mapping $P^1$ to itself and depending holomorphically on a parameter $t$ ranging over some complex manifold, and the bifurcation set (or locus) $B(f)$ is the set of $t$ at which the dynamics of $f_t$ undergo a discontinuous change. For a positive integer $d\geq 2$, $M_d$ is the set of $t$ for which the function $z\mapsto z^d+t$ has a connected Julia set, and $\partial M_d$, the boundary of $M_d$, is the bifurcation locus for the family $\{z\mapsto z^d+t\}_t$.)

Even without knowing exactly what the "category of bifurcations" is, I can see an analogy between the claim about Mandelbrot sets and the concept of initial objects, but presumably something more definite is intended. My question is thus how to interpret the second sentence quoted from McMullen's paper, or more simply: what is the "category of bifurcations"?

  • $\begingroup$ There's something fishy here with the "or": are there supposed to be a family of nonisomorphic initial objects? $\endgroup$ – Qiaochu Yuan Jan 19 '10 at 3:39
  • $\begingroup$ On the face of it, that seems to be what it's saying, but that would be impossible. I'm hoping the proper definition of the category will clear that up. $\endgroup$ – Darsh Ranjan Jan 19 '10 at 4:38
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    $\begingroup$ It might be that rather than a single initial object, there is a "multi-initial-object": ncatlab.org/nlab/show/multilimit . For example, the category of fields has a multi-initial-object, namely the family of all prime fields. $\endgroup$ – Mike Shulman Jan 19 '10 at 6:19
  • $\begingroup$ I just thought of that example after I posted that comment, but I wasn't sure if it had a name. Thanks! $\endgroup$ – Qiaochu Yuan Jan 19 '10 at 14:27
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    $\begingroup$ I'm sure that "the category of bifurcations" is a (the unique?) category whose objects are bifurcation loci, but I'm not sure what the morphisms are. I happen to know that the bifurcation locus of the family $f_t(z) = z - p_t(z)/p'_t(z)$, where $p_t(z) = (z^2 - 1)(z - t)$, is composed of many rotated and scaled copies of $\partial M_2$, so the morphisms ought to be the sort of thing that can take $\partial M_2$ to many (I think infinitely many) scaled and rotated copies of $\partial M_2$. From other context in the paper I think the copies must be allowed to be quasi-conformal images. $\endgroup$ – Aaron Golden Nov 24 '12 at 17:05

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