The operad of topological simplices, which I'll denote $\Delta$, has as $n$-ary operations the set $$ \Delta_n:=\left\{P\colon\{1,\ldots,n\}\to[0,1]\;\middle|\;1=\sum_{i=1}^n P(i)\right\} $$ of probability distributions on an $n$-element set; composition is by "weighted sum". This operad is $\Sigma$-free, meaning that the symmetric group $\Sigma_n$ acts freely on $\Delta_n$.

A $\Delta$-algebra consists of a pair $(X,h)$ where $X:\mathbf{Set}$ is a set and $h\colon\sum_{n:\mathbb{N}}\Delta_n\times X^n\to X$ is a function, satisfying the usual identity and composition laws.

Every convex set is an $\Delta$-algebra. One might be tempted to think that these are precisely the $\Delta$-algebras, except for the fact that every monoid (i.e. every algebra of the terminal $\Sigma$-free operad) is also a $\Delta$-algebra. In terms of $\Delta$-algebras, monoids discard the probabilistic weighting of each element and instead just react to the list of elements; that is, $h$ factors through $\sum_{n:\mathbb{N}}X^n\to X$. On the other hand, convex sets discard the 0-weighted elements.

Question: How can we characterize the $\Delta$-algebras?

The operad of topological simplices, which I'll denote $\Delta$, has as $n$-ary operations the set $$ \Delta_n:=\left\{P\colon\{1,\ldots,n\}\to[0,1]\;\middle|\;1=\sum_{i=1}^n P(i)\right\} $$ of probability distributions on an $n$-element set; composition is by "weighted sum".

A $\Delta$-algebra consists of a pair $(X,h)$ where $X:\mathbf{Set}$ is a set and $h\colon\sum_{n:\mathbb{N}}\Delta_n\times X^n\to X$ is a function, satisfying the usual identity and composition laws.

Every convex set is an $\Delta$-algebra. One might be tempted to think that these are precisely the $\Delta$-algebras, except for the fact that every monoid (i.e. every algebra of the terminal plain operad) is also a $\Delta$-algebra. In terms of $\Delta$-algebras, monoids discard the probabilistic weighting of each element and instead just react to the list of elements; that is, $h$ factors through $\sum_{n:\mathbb{N}}X^n\to X$. On the other hand, convex sets discard the 0-weighted elements.

Question: How can we characterize the $\Delta$-algebras?