Distribution of a pi-finite Anima

Let $X$ be a $\pi$-finite space. We construct a map of sets

$${\rho }_X:{\pi }_0(X)\to \mathbb{Q}$$ $$x\mapsto {\rho }_X(x):=1/|{\pi }_1(X,x)|$$

We’d like to normalize this: summing over $x$ is

$$c_X:=\sum _{x\in {\pi }_0(X)}\frac{1}{{\pi }_1(X,x)}$$

And we can define a normalized distribution by

$$P_X:=\frac{1}{c_X}{\rho }_X$$

In words, we’ll refer to the quantities defined here as

• ${\rho }_X$: unormalized distribution for $X$
• $c_X$: scale of $X$
• $P_X$: (normalized) distribution for $X$

Picture:

Consider the case that $X$ is 1-truncated: so a finite “ordinary” groupoid. Let’s picture this as following physical objects:

• a connected component is some shape stiched out of rubber bands. Every band in this ball has the same color labelling the component.
• the entirety of the groupoid then loooks like a finite collection (${\pi }_0(X)$) of such rubber band creatures, each with a different color.

Now we throw these rubber animals into a sack, and ask a blindfolded participant to reach in their pinky, and hook exactly one rubber band segment. Our constructions precisely describe the statistics of what color the participant will end up pulling out.

In other words, the probability distribution measures relationships (rubber bands) rather than objects (intersection points of rubber bands).

Generalization

In general, an $\infty$-groupoid will be stiched out not of rubber bands, but also of 2-dimensional sheets, 3-dimensional volumes, 4-dimensional volumes, all the way up. $X$ will be a bunch of such higher-dimensional rubber creatures. Since $X$ is $\pi$-finite, there’s a finite number of these cells, so we can in fact label all such cells in $X$ in a finite amount of time.

The set up is the following

• Starting with $X$ above, throw the set of such cell labels

Action items

• Do some sample calculations:
  • Boltzman Entropy of this distribution
• Incorporate higher-homotopy corrections from the $\infty$-groupoid cardinality.
• Infinite-symmetries (e.g., $T^n$, $S^n$) , via using the chromatic tower to extrapolate, as Lior does.