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](https://en.wikipedia.org/wiki/Quasi-arithmetic_mean).


**Background:**
The operad $(P,\circ)$ features explicitly in Leinster's [operadic characterization of entropy](https://www.ncatlab.org/johnbaez/show/Entropy+as+a+functor), 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](https://ncatlab.org/nlab/show/symmetric+sequence), though you might also call it a topological species). An [operad](https://ncatlab.org/nlab/show/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 $P$.



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


