Biequivalence between graded distributive Markov categories and commutative affine graded relative monads

Establish a biequivalence between the 2-category of graded distributive Markov categories and the 2-category of commutative affine graded relative monads (over distributive categories), demonstrating that the constructions sending a graded commutative affine relative monad to its Kleisli graded Markov category and, conversely, sending a graded Markov category to its associated graded relative monad, are inverse up to biequivalence.

Background

The paper develops graded versions of Markov categories and relative monads, and provides constructions connecting them in both directions: from a graded commutative affine relative monad to a graded Markov category via a Kleisli construction, and conversely from a graded Markov category to a graded relative monad via a presheaf-based construction.

Although these constructions are presented, the authors stop short of proving that they yield a full biequivalence, leaving the categorical correspondence between these frameworks as a conjecture. Establishing such a biequivalence would canonize the relationship and unify the semantics of imprecise probability across both categorical formulations.