How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?

Question

Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by multiplication. That is,

• Let $p \in \Delta^{n-1}$ be given in coordinates by $p = (p^1,\dots,p^n)$.

• For each $i=1,\dots,n$, let $p_i \in \Delta^{n_i-1}$ be given in coordinates by $p_i = (p_i^1,\dots,p_i^{n_i})$.

Then the operad composition is given by

$p \circ (p_1,\dots,p_n) = (p^1 p_1^1, p^1 p_1^2,\dots, p^1 p_1^{n_1}, p_2 p_2^1, \dots, p^n p_n^1, \dots, p^n p_n^{n_n})$.

For example, $(1/3,2/3) \circ ((1),(1/4,1/4,1/2)) = (1/3,1/6,1/6,1/3)$.

(I apologize for the notation -- in particular the tuples on the two sides of the above equation mean different things -- on the left, it's a list of inputs to the $\circ$ operation, while on the right it's a list of coordinates. Another way to write this would be $(p^i)_{i=1}^n \circ ((p_1^j)_{j=1}^{n_1},\dots, (p_n^j)_{j=1}^{n_n}) = (p^i p_i^j)_{1 \leq i \leq n, 1 \leq j \leq n_i}$.)

We can twist this operad by any convex increasing homeomorphism $f: [0,1] \to [0,1]$ as follows. Define $f_\ast: \Delta^{n-1} \to \Delta^{n-1}$ by $f_\ast(x_1,\dots,x_n) = (\frac{f(x_1)}{Z},\dots, \frac{f(x_n)}{Z})$ where $Z = f(x_1) + \dots + f(x_n)$; this is a $\Sigma_n$-equivariant homemormophism ( invertibility requires $f$ to be convex ), and then set

$p \circ^f (p_1,\dots, p_n) = f_\ast^{-1}(f_\ast(p) \circ (f_\ast(p_1),\dots, f_\ast(p_n)))$

Question: Is every topological operad structure on the symmetric sequence $P$ of the form $\circ^f$ for some convex increasing homeomorphism $f: [0,1] \to [0,1]$?

Guessing the answer is "yes" is motivated by the characterization of quasi-arithmetic means.

Background: The operad $(P,\circ)$ features explicitly in Leinster's operadic characterization of entropy, and twists $(P,\circ^f)$ for certain $f$ are used by Baez, Fritz, and Leinster to characterize certain deformations of entropy. Certainly these operads are implicit in countless mathematical pursuits.

Explicitly: The fixed data is the sequence of topological spaces $(P_0,P_1,\dots)$ and the $\Sigma_n$ action on $P_n$ (this is called a symmetric sequence, though you might also call it a topological species). An operad structure on the symmetric sequence $P$ is a monoid structure for the substitution monoidal product. That is, it consists of continuous operations $\circ: P_n \times P_{n_1} \times \dots \times P_{n_n} \to P_{n_1 + \dots + n_n}$ for each $n,n_1,\dots,n_n \in \mathbb N$ satisfying:

• unitality: $1 \circ p = p = p \circ (1, \dots, 1)$ for some $1 \in P_1$

• associativity: The two ways of associating $p \circ (p_1,\dots,p_n) \circ (p_{1,1},\dots, p_{n,n_n})$ are equal.

• $\Sigma_n$-equivariance: $(\sigma \cdot p) \circ (p_1,\dots, p_n) = \sigma' \cdot (p \circ (p_{\sigma^{-1}1},\dots, p_{\sigma^{-1}n}))$ for each $\sigma \in \Sigma_n$, where $\sigma' \in \Sigma_{n_1 + \dots + n_n}$ is the image of $\sigma$ under the natural inclusion $\Sigma_n \ltimes (\Sigma_{n_1} \times \dots \times \Sigma_{n_n}) \hookrightarrow \Sigma_{n_1+\dots+n_n}$.

The question asks to classify operad structures on the symmetric sequence $P$.

Answer 1

Abusing notation, write m for the uniform distribution on each finite set and define p o p' = m for any p, p' not equal to the identity.

This is possibly a limit of operads of the formula you give where you choose homeomorphisms which tend (pointwise) to a constant function f(x) = c. But I think qualifies as an operad structure not in your suggested classification.

Answer 2

Put $Q_n=\{(x\in [0,1]^n:\text{max}(x_1,\dotsc,x_n)=1\}$. Then there is an isomorphism $Q\to P$ of symmetric sequences given by $x\mapsto x/\sum_ix_i$. Define $$ p \circ (p_1,\dots,p_n) = (\min(p^1,p_1^1), \min(p^1,p_1^2),\dots, \min(p^1 ,p_1^{n_1}),\min(p_2,p_2^1), \dots,\min(p^n,p_n^1), \dots,\min(p^n,p_n^{n_n})). $$ This gives an operad structure on $Q$. Let $C$ be the commutative operad, so that $C_n$ is a singleton for all $n$. There is an operad morphism $C\to Q$ sending the unique point of $C_n$ to $(1,\dotsc,1)$, but there is no operad morphism $C\to (P,\circ^f)$, so $Q\not\simeq(P,\circ^f)$ as operads.

Answer 3

I just want to verify that

Fact: If $f: [0,1] \to [0,1]$ is any increasing homeomorphism, then

$$f_\ast: \Delta^{n-1} \to \Delta^{n-1}, (x_i)_i \mapsto (f(x_i)/(f(x_1) + \dots + f(x_n)))_i$$

is a ($\Sigma_n$-equivariant) homeomorphism.

It is not necessary for $f$ to be convex as I had initially asserted.

Proof: Let $\vec y = (y_i)_i \in \Delta^{n-1}$, and assume without loss of generality that $y_1 = \max_i y_i$ (and in particular $y_1 \neq 0$). Because $f_\ast$ is a continuous map between compact Hausdorff spaces, it suffices to show that $\vec y$ has a unique inverse under $f_\ast$.

Extend $f_\ast$ to a map $[0,1]^n \setminus (0,\dots,0) \to \Delta^{n-1}$ by using the same formula as above. Note that for $\vec x \in [0,1]^n$, we have $f_\ast(x_1,\dots,x_n) = (y_1,\dots,y_n)$ if and only if $x_1 \neq 0$ and $f(x_i)/f(x_1) = y_i/y_1$ for all $i$. That is, defining $\nu: (0,1] \to [0,1]^n$ by $\nu_i(x) = f^{-1}(y_i f(x_1)/y_1)$, we have $f_\ast(\vec x) = \vec y$ if and only if $x_1 \neq 0$ and $\vec x = \nu(x_1)$. (The formula for $\nu$ makes sense because $y_i \leq y_1$ and $f(x_1) \leq 1$).

We now claim that there is a unique $x \in (0,1]$ such that $\nu(x) \in \Delta^{n-1}$, i.e. such that $\sum_i \nu_i(x) = 1$, which is equivalent to our claim that $\vec y$ has a unique inverse under $f_\ast: \Delta^{n-1} \to \Delta^{n-1}$. This holds because the function $(0,1] \to [0,\infty), x \mapsto \sum_i \nu_i(x)$ is strictly increasing and continuous, tending to 0 as $x \to 0$, and with a maximum value at $x=1$ of $\nu_1(1) + \nu_2(1)+ \dots + \nu_n(1) = 1 + \nu_2(1) + \dots + \nu_n(1) \geq 1$. So by the intermediate value theorem, this function attains the value 1, and because it is strictly increasing it attains this value for a unique $x$.