The n-Category Café

Did you see this question at Math Overflow, which asks for an interpretation of the claim that Mandelbrot sets are initial objects in the category of bifurcations?

If nothing else, I learned about the notion of multilimit from Mike Shulman’s comment as people there tried to interpret the claim.

Are you going to comment on Erik Verlinde’s provocative new paper some week soon? It’s starting to spawn interesting followups.

I found a video where one can make IFS fractals, like Sierpinski or fern-like using 3 projector and a camera, WITHOUT the use of a computer. All this because of the feedback of images. There is a cool music going on while someone generates and changes the fractals.

Mobius strips are cool, and the Crab Canon is cool, but they’re essentially different. Notice that in the video, the two players are still going around the Mobius strip in opposite directions, and each is keeping to its own side of the strip. Moreover, in spite of visually putting in a twist, the “backwards” player is really playing the sound in a mirror, not up-side-down. There’s a reason Bach calls it “crab”: it can be played forward and backward.

Thus, the correct visualization is not a Mobius strip at all, but the orbifold with boundary formed by reflecting the rectangle in half. Making this is easy: take a piece of paper with the music written on one side, and fold it so that the music is on the outside. In this way, each side of the orbifold has half the music on it. Now start at the non-mirror end, but play both sides, reflecting through the orbifold boundary and continuing until you’re back where you started.

Thanks for pointing our Carlos Simpson’s book Homotopy Theory of Higher Categories!

Am reading…

Some random comments that came to mind during my first rough go through chapter I, “History and motivation”.

• I was looking for mentioning of Charles Rezk’s nice stuff on Theta-spaces, which claims to achieve much of what this book is about. From the remark on p. 23 I gather this got in a bit too late to be treated more in detail in this book. (?)

• page 24: wondering about the conjunction by which $n$-stacks and $n$-gerbes are mentioned here. Larry Breen’s work on 1- and 2-gerbes concerns 1-groupoid and 2-groupoid valued sheaves, both examples of $(n,0)$-sheaves that are fully under control for all $n \leq \infty$ (and, in hindsight, have been since a couple of decades). Maybe I am misreading the notation. I think the next interesting application of $(\infty,2)$-sheaves is not $n$-gerbes which are a special case of principal $\infty-bundles$ but of quasicoherent $(\infty,2)$-sheaves corresponding to $\infty$-vector bundles (as principal bundles form a groupoid, vector bundles a category, and so on).

• page 25, am enjoying seeing nonabelian $\infty$-stack cohomology amplified.

  But I am wondering if we can’t make confidently stronger statements by now: as we recently discussed here, feeding Rezk’s model structures on $(\infty,n)$-categories into Barwick’s result on enriched Bousfield localization of the presheaf model categories at the set of Cech-cover-nerves should give the $(\infty,1)$-categories of non-hypercomplete $(\infty,n)$-stacks for all $n$ (and probably even the corresponding $(\infty,n+1)$-categories/Theta-spaces). What would be missing were “just” the intrinsic point of view, where we’d be able to justify the Bousfield localization step concretely as modelling a left exact localization intrinsic to $(\infty,n)$-categories, as Lurie did for $n=1$ in his book for Dugger’s theory.

• still page 25: now fundamental $\infty$-groupoids aka Poincaré-groupoids and their relation to and motivation from the vanKampen theorem. Readers here know that I spent some time thinking about this. In the end I felt that the construction of the path $\infty$-groupoid functor on lined $\infty$-stack $\infty$-toposes that is left Quillen and preserves weak equivalences between fibrant objects is pretty much the answer, as this respect for weak equivalences is the desired vanKampen codescent, really.

  So I am skipping ahead to section 3.6 “Poncaré $n$-groupoids” but probably I need to read more later on. (?)

• page 26: that “$\mathcal{F}$” is probably meant to read “$A$” (?)

• page 27 “gerbs” – one has to be careful with this typo, a student of Hitchin once introduced this (i.e. gerbe without one of the e-s) as a technical term! :-)

So much for the moment. Any comments on my comments?

It is a bit of an urban legend that Bach improvised a six-voice fugue. There is certainly a six-voice fugue in the musical offering, but someone once compared the feat of improvising one to playing many parallel games of blindfold chess. I think the concensus (talking from memory here - a reference would be nice) is that Bach improvised a thee-voice fugue for the king, and included it (or one like it) in the offering.

Conc. Bach: Historians of music apparently suspect that C. Ph. E. Bach suppressed the last parts of J.S. Bach’s “Art of Fugue”, here is an interview with Ton Koopman mentioning that.

Particle physicists should look at the above diagram and think about how Feynman diagrams with closed loops in them lead to infinities. Category theorists should think about “traces” and how sometimes traces diverge. It is my job to make these analogies precise. But not today.

For some reason, this reminded me of St. Augustine. “Oh, Lord, grant me precision and rigour — but not yet!”

Hi John! About your question on Mandelbrot and Julia set similarity:

The idea is the following: consider a parameter $c$; then it belongs to the Mandelbrot set iff $$((c^2+c)^2+c)^2 +...$$ is bounded. Now consider the corresponding *filled in* Julia set, namely those $z$ such that the orbit of $z$ under $f(z)=z^2+c$ is bounded. To any point $c$ in the Mandelbrot set corresponds the point $z=c$ such that $c$ is coloured “black” iff $z$ is coloured black. Then, with the help of lots of $\epsilon$’s and $\delta$’s, you can see that microscopically changing $c$ has roughly a comparable effect as microscopically changing $z$.

I’m not aware of statements that are true everywhere, but mainly of conjectures as to how these things behave. The fundamental conjecture (from my perspective) is that the Julia and Mandelbrot sets are described by simple combinatorial models: the boundary of the Mandelbrot and connected Julia sets are quotients of a circle by a simple equivalence relation. Remember that the Julia set of $z^2+c$ is connected iff $c$ belongs to the Mandelbrot set.

To understand all of that, note that $z^2+c$ is conjugate, in the neighbourhood of $\infty$, to the map $z^2$; i.e. there exists a map $B:D\to\widehat C$ where $D$ is the open unit disk, such that $B(0)=\infty$ and $B(z^2)=f(B(z))$; and $B'(0)=1$ for normalization. The boundary of the Julia set “should be” the image of the boundary of $D$, but it’s only known for some values of $c$. Consider radial lines in $D$; their images are rays $R_\theta$ in the complement of the Julia set, starting at $\infty$, and conjecturally converging. Points in the boundary of the Julia set are therefore parameterized by an angle, with different angles possibly describing the same point.

Now the complement of the Mandelbrot set may also be parameterized by a disk; namely, for $c\not\in$ the Mandelbrot set, set $f(z)=z^2+c$ and consider $G(c)=\lim 2^{-n}\log f^n(0)$. Then this defines a holomorphic map $G$ (careful with the log!), and again points on the boundary of the Mandelbrot set get an angle-valued parameter. $G$-preimages of radial lines are called “parameter rays” $P_\theta$.

Another link between what happens in Mandelbrot and Julia space can now be described. Consider pairs of rays in Julia space that land at the same boundary point. A pair is called “characteristic” if it separates $0$ from $c$ and none of its forward images separates $c$ from the initial pair. Then if the rays $R_\theta,R_\phi$ form a characteristic pair, then $P_\theta$ and $P_\phi$ land at the same point in the boundary of the Mandelbrot set.

It follows that the “pinching” that describes the boundary of the Mandelbrot set as a quotient of a circle is well related to the “pinching” that describes Julia sets.

I was a bit bemused to see that particular Julia set so neatly approximated by a speck of the M-set $M$, particularly because it’s so disconnected — recent reading calls it a species of “Fatou Dust” — which means it corresponds to a point very much NOT in the M-set; so I asked gnofract4d to show me the Julia for the point identified, and it looked nothing like that! Trying to zoom in the Mandelbrot disappointingly pixelated before I found anything similar, but I also suspect a little distortion due to rounding — I’m not good at estimating these sorts of error, you know. But then reading the wikipedia article seemed to clear things up for me (that article’s captions will have to be edited, because they really are misleading!)

These “embedded Julia sets” aparently aren’t part of the classic “easy” renormalization; instead they seem to show up somewhat in analogy to the satelite sub-mandelbrot structures;

Disclaming honestly, this is for myself some mildly wild conjecture: for a sub-mandelbrot, it looks like there’s a neighborhood $U$ of $M$ and an injective conformal map $\varphi:U\rightarrow U$ which is in some sense locally maximal such that $\varphi|M\rightarrow M$ — and furthermore, $M\backslash \varphi(M)$ accumulates at $\partial \varphi(M)$. The fine structure of this accumulation is quite intriguing: suppose the local filigree looks like X; then approaching the satelite mandelbrot, there’s a progression to finer spacing of parallel shorter branches of rough shape X. It’s something akin to the fuzzy appearance of the logistic Feigenbaum tree just prior to the emergence of an attracting odd cycle (which surely isn’t a coicidence). However, approaching the boundary of $\varphi(M)$, experimentally there are other regions than second-order satelites where X-shaped branches can accumulate in this manner, and if there is such an exotic accumulation near a point $\varphi(c)$, then it looks like the Julia set for $c$.

It should then also follow that these embedded Julias are internally Cantor, because of necessity they correspond (via $\varphi$) to points $c$ outside $M$.

Mixo Lydian sent me an email answering my question about why Japan has currents of two different frequencies — 50 and 60 cycles per second. As expected, there’s some history involved:

The 50Hz/60Hz divide in Japan is due to historic reasons. Towards the end of the Meiji era, Japan made the switch from DC to AC. Tokyo Dento (Japan’s first electric power company) adopted 50Hz German AEG generators while its rival Osaka Dento decided to adopt 60Hz American GE generators to power their respective electric grids.
Neighboring regions built their electric infrastructure adopting either Tokyo or Osaka standards which has led to a east-west / Tokyo-Osaka divide which continues to the present day, the exact border being the Fuji river which runs thru Shizuoka prefecture: east of the river the frequency is 50Hz, west of river the frequency is 60Hz.

This has hilarious consequences for the town of Shibakawa-cho, Shizuoka. The Fuji river runs directly thru Shibakawa-cho: some parts of town use 50Hz while others use 60Hz! All you have to do is cross a bridge to alternate between (intentional pun)!

Hope this has been helpful.

From Fractal Groups to Fractal Sets
Authors: Laurent Bartholdi, Rostislav I. Grigorchuk, Volodymyr V. Nekrashevych

“In its survey, the paper discusses finite-state transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems,Hecke-type operators, C^*-algebras, random matrices, ergodic theorems and entropy of non-commuting transformations. Self-similar groups appear then as a natural weaving thread through these seemingly different topics.

The aim of this paper is to present a survey of ideas, notions and results that are connected to self-similarity of groups, semigroups and their actions; and to relate them to the above-mentioned classical objects. Besides that, our aim is to exhibit new connections of groups and semigroups with fractals objects, in particular with Julia sets.”

Wikipedia has an article on the Newton fractal which may relate J- to M-sets.