What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by

$$ DX = \left\{ p \in [0,1]^X \ \Big|\ \left|p^{-1}(0, 1]\right| \le \aleph_0,\ \sum_{x \in X} p(x) = 1 \right\}? $$

I can only find information on the finitary version, e.g. here.


The algebras of this monad and closely related ones have been introduced by Pumplün and Röhrl. For example the introduction of a paper by Börger and Kemper provides a good summary,

Pumplün and Röhrl [6] introduced the notion of totally convex space. These spaces are the Eilenberg-Moore-algebras for the monad induced by the unit-ball functor $\bigcirc:\mathbf{Ban}_1\to \mathbf{Set}$, where $\mathbf{Ban}_1$ is the category of Banach spaces and linear operations of norm $\leq 1$. They can be described as sets with formal infinitary linear combination maps $X^\mathbb{N}\to X,\: (x_i)_{i\in\mathbb{N}} \mapsto \sum_{i\in\mathbb{N}}\alpha_i x_i$, where $\alpha_i\in K$, $\sum_{i\in\mathbb{N}}|\alpha_i|\leq 1$, $K\in\{\mathbb{R},\mathbb{C}\}$, subject to two axioms.

Pumplün and Röhrl also considered the categories obtained by restricting the operations to positive coefficients (maybe even with $\sum_{i\in\mathbb{N}}\alpha_i=1$) and/or finite arity ([4,5,8]). [..]

The relevant reference here is [4], which points to Pumplün's paper "Banach spaces and superconvex modules". The latter seem to be the algebras of your monad, as are the "superconvex spaces" of the paper linked to above. (Although there is a cryptic remark on whether the empty set counts as superconvex or not.) I haven't been able to track down reference [4], but a web search for "superconvex module" and "superconvex space" reveals some other papers.

Finally, it may be worth emphasizing that your monad probably arises from a very natural adjunction similar to the one formed by the unit ball functor $\bigcirc:\mathbf{Ban}_1\to\mathbf{Set}$ and its left adjoint. I haven't checked the details, but I bet that it is the monad induced by the forgetful functor from the category of norm-complete base norm spaces with base-preserving positive linear maps as morphisms to $\mathbf{Set}$. Maybe you already know?


The algebras for this monad can be described in essentially the same way: they are sets in which it makes sense to to take "convex combinations" of countably many elements. More precisely, an algebra is a set $X$ together with an operation $$((a_0, a_1, a_2, \ldots), (x_0, x_1, x_2, \ldots)) \mapsto \sum_{i = 0}^\infty a_i x_i$$ defined for every sequence $(a_0, a_1, a_2, \ldots)$ of non-negative real numbers such that $\sum_{i = 0}^\infty a_i = 1$ and every sequence $(x_0, x_1, x_2, \ldots)$ of elements of $X$, such that $$\sum_{i = 0}^\infty a_i \left( \sum_{j = 0}^\infty b_{i,j} x_{i,j} \right) = \sum_{k = 0}^\infty \left( a_{p(k)} b_{p(k), q(k)} \right) x_{p(k), q(k)}$$ where $k \mapsto (p(k), q(k))$ is your favourite bijection $\mathbb{N} \to \mathbb{N} \times \mathbb{N}$, and $$\sum_{i = 0}^\infty a_i x_i = \sum_{j = 0}^\infty b_j y_j$$ whenever $\sum_{i : x_i = z} a_i = \sum_{j : y_j = z} b_j$ for all $z \in X$.

The most concise way of giving the complete definition is to use the monad in your question.


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