The n-Category Café

The Great Blue Hole

Congrats Tom and sorry for preempting you with the other post. I just couldn’t contain my enthusiasm!

Many congratulations, Tom. I had to teach the Yoneda lemma today and looked up your “Basic Category Theory”. The start of Chapter 4 made my day: “A category is a world of objects, all looking at one another. Each sees the world from a different viewpoint”.

I look forward to read equally inspiring explanations in the new book!

Beautiful book Tom! 400 plus pages, but it is so nicely organized and readable that I felt immediately drawn in rather than intimidated. It probably helps that I’ve followed many of the posts here about the papers that the book covers, but everything I’ve read so far just flows as if the reader could fill in the next sentence on their own. That means a ton of thought and background went into every sentence.

Okay, so now I have a question!

I’m looking at the $q$-logarithmic entropy and the $q$-logarithmic information loss, specifically for $q=2$. It looks like $L_2(f)$ can be thought of as electrical resistance, and as such obeys Ohm’s law.

For instance, for parallel processes $f,f'$ let $L_2(f) =a$ and $L_2(f') =b.$ Now we are given the convex linearity of $L_2$, but I want to choose a specific $\lambda$ for these parallel processes. I choose $\lambda = \frac{1/a}{1/a+1/b} = \frac{b}{a+b}.$ Now the axiom says that $L_2$ of the weighted parallel processes is $\lambda^2 L_2(f)+(1-\lambda)^2 L_2(f').$ Using my choice of $\lambda$, that’s equal to $\frac{a b}{a+b}.$

…which is exactly the formula for the effective resistance of parallel electrical resistors that obey Ohm’s law. We can describe the $\lambda$ chosen above as the ratio of the current through $f$ to the total current through $f$ and $f'$, where voltage is 1. Therefore, for any given connected collection of morphisms, the information loss as measured by $L_2$ could be modeled with wires in an electric circuit.

So my question is: is this a well-known fact, or a simple example of a bigger well-known theorem?

Nope, nope. Answering my own question here…

If electrical resistance is a functor that obeys the 2-convex-linearity equation for a special value of $\lambda$ only, it does not deserve to be called 2-convex-linear.

What we have are a two different continuous functors, $L_2$ and $R$ (resistance), both with the non-negative reals as codomain. $L_2$ obeys a stronger condition than $R$. The 2-convex-linearity condition implies the condition for Ohmic resistance, for parallel paths in the circuit. I’m not sure how to state this in a slogan!

A little more about why this might be nice to know: if you have a (multi)graph with resistance values assigned to its edges and a selection of vertices as the boundary nodes (terminals) then there are some well-known graph transformations that leave the measured pairwise resistances at the terminals invariant. One is the Y-$\Delta$ transformation shown in the linked preprint above, another is replacing multi-edges with their equivalent single edge.

This looks like a very nice book! I suspect I will be recommending it as supplemental reading at least the next time I teach statistical physics. (I recommended a lot of supplemental texts when I taught that this past spring, because the class wasn’t able to meet in person after March…) It calls to mind a good many “sweet sorrows” — ideas that I found quite engaging and worked with for a longish while but was never able to get into a satisfying shape. For example, I had the notion that in a similarity-weighted diversity index

$$D_2^Z(p) := \left(\sum_{i j} Z_{i j} p_i p_j\right)^{-1} ,$$

the sum over $i$ and $j$ is the expected score in a game whose goal is agreement and the two players play randomly. This leads to the question of what happens when the game involves more than two players — what about contractions of the form $Z_{i j k} p_i p_j p_k$? These arise in quantum information, where the $Z_{i j k}$ are the real parts of certain geometric phases, but we could also run into three-party similarity measures when comparing species. To take an artificial but pretty example, suppose $\{t_1,t_2,\ldots,t_7\}$ are seven different phenotypic traits, that $\{s_1,s_2,\ldots,s_7\}$ are seven species, and that traits and species correspond to points and lines in the Fano plane. Each species has three of the seven traits, every two species have one trait in common, and every trait is found in three species. All pairs are alike; we could say that $Z_{i j} = \frac{1}{3}$ for all $i \neq j$. But not all triads are alike, because a set of three lines in the Fano plane can either meet at a common point or not. Some sets of three species are more similar than others.

To move a little in the direction of biology, we could take a phylogenetic network. Let $S$ be a set of ancestral species and $T$ be a set of descendant species, with a directed graph $G$ providing paths from $S$ to $T$. These paths might diverge if there is an evolutionary radiation, and they might converge if there is hybridization or horizontal gene transfer. This structure defines a matroid on $T$, whose rank function $r(U)$ for $U \subset T$ is the size of the smallest set of vertices having the property that all paths from $S$ to $U$ must pass through it. (This kind of matroid is known as a gammoid.) The rank function $r$ is a kind of dissimilarity measure for the descendant species. A subset $U \subset T$ is independent in the matroid-theoretic sense if there exists a set of vertex-disjoint paths from $S$ whose ending points are exactly $U$; in biological language, this would say that the species in $U$ do not have a genetic common ancestor. (At least, to find one, you’d have to go back further into the past than $S$.)

The funny thing is that rank functions of matroids behave a lot like Shannon information of sets of random variables. They always satisfy $r(U) \leq r(V)$ for $U \subseteq V$, and they are submodular or strongly subadditive: for all $U, V$,

$$r(U \cup V) + r(U \cap V) \leq r(U) + r(V).$$

So, we have an entropy-like quantity coming just out of the graph structure, even before we put a probability distribution on the species. I find that a bit odd!

This leads into the topic of higher-order mutual information defined by inclusion-exclusion, as you do in Remark 8.1.11, and how to make sense of it.