On the Probabilistic Convergence Spaces: Monad and its Eilenberg–Moore Category

Motivated by the category of probabilistic convergence spaces — a supercategory of the category of topological spaces; recently, we brought to light the categories of probabilistic convergence groups, probabilistic metric probabilistic convergence groups, probabilistic convergence transformation groups, along with their underpinning natural examples. The purpose of this paper is, first, to establish a result on the isomorphism between the categories of probabilistic metric groups, and probabilistic metric probabilistic convergence groups. Second, among others, we explore a monad in relation with probabilistic convergence groups, and probabilistic convergence spaces, and their related algebras. In so doing, we consider a product of the categories of probabilistic convergence groups and probabilistic convergence spaces in an attempt to construct a monad on it such that the corresponding category of algebras, the so-called Eilenberg–Moore category, is isomorphic to the category of probabilistic convergence transformation groups. Finally, invoking so-called Beck’s theorem on characterization of algebras, and starting with a particular adjunction, we achieve a monad. Conversely, given a monad, we obtain an adjunction which coincides with the original monad.