Characterize algebras of the "topological simplices" operad

Question

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?

Answer 1

One thing to notice is that every element in $\Delta_n$, for $n \geq 2$, is obtained by iterated composition of elements of $\Delta_2$ (these are parametrized by elements $t$ in the unit interval). In other words, the binary operations for this theory generate all the operations, and in a specific way: for a model or algebra $X$, letting $\theta$ be any element of $\Delta_n$, and denoting the operation associated with $\theta$ by $m_\theta: X^n \to X$, there exist elements $t_1, \ldots, t_{n-1} \in \Delta_2$ such that (associating to the right)

$$m_\theta(x_1, \ldots, x_n) = m_{t_1}(x_1, m_{t_2}(x_2, \ldots m_{t_{n-1}}(x_{n-1}, x_n)\ldots )) \qquad (1)$$ (Pause for a moment to consider the example of convex sets. Here $m_t(x, y) = tx + (1-t)y$.) For $\theta = (\theta_1, \ldots, \theta_n)$ in the interior of the simplex $\Delta_n$, the $t_i \in \Delta_2$ are uniquely determined by the formulas

$$t_1 = \theta_1, \qquad t_i = \frac{\theta_i}{1 - \theta_1 - \ldots - \theta_{i-1}}$$ for $1 < i < n$. But regarding points $\theta$ on the boundary of $\Delta_n$: as soon as $t_i = 1$, the remaining parameters $t_{i+1}, t_{i+2}, \ldots, t_{n-1}$ are no longer uniquely determined; indeed, for any choice of those parameters past $t_i = 1$, equation (1) holds. Put differently,

$$m_1(x, m_t(y, z)) = m_1(m_1(x, y), z) \qquad (2)$$ is an equation that holds universally in $\Delta$-algebras, and it follows that

$$m_1(x, m_s(y, z)) = m_1(x, m_t(y, z))$$ is an identity that holds for all $s, t \in \Delta_2$.

All of this holds in particular for operations $m_\theta$ that are derived from binary operations $m_t$ by iterated composition, sometimes associating to the left. In other words, we have equations

$$m_s(m_t(x, y), z) = m_{st}(x, m_{\frac{s-st}{1-st}}(y, z)) \qquad (3)$$ (provided that $st \neq 1$) which play the role of associativity equations. This may look strange at first, but for the example of convex algebras, it's a simple calculation based on rearranging convex combinations:

$$m_s(m_t(x, y), z) = s(tx + (1-t)y) + (1-s)z = stx + (1-st)\left(\frac{s-st}{1-st} y + \frac{1-s}{1-st} z\right).$$

So, we may characterize $\Delta$-algebras as sets $X$ equipped with binary operations $m_t$, with $t$ ranging over $[0, 1]$, subject to the associativity conditions (2) and (3) above.

(We could even use equations (2) and (3) to set up a rewrite system for derived iterated compositions of binary operations, as tracked by planar binary trees, where in each application of of these cases we rewrite the left side of the equation to the right side.)

There are a number of subsidiary consequences of (3). For example, mirroring equations of type (2), we have the equation

$$m_0(m_t(x, y), z) = m_0(x, m_0(y, z)). \qquad (4)$$ So $m_0, m_1$ are associative each in their own right, and equations (2) and (4) state their "absorption" properties, and there are some other mild curiosities like

$$m_s(m_0(x, y), z) = m_0(x, m_s(y, z)), \qquad m_1(m_t(x, y), z) = m_t(x, m_1(y, z)).$$