Eilenberg-Moore Algebras for Stochastic Relations
Ernst-Erich Doberkat · 2007
It is shown at the end of Section 1.6.3 that the adjunction constructed from the Eilenberg-Moore algebras and the one constructed through the Kleisli category form in some sense the extreme points in a category of all adjunctions from which the given monad can be recovered. From this, the algebraic interest to identify these algebras is derived. The algebras for the power set monad (dubbed here the Manes monad) are well known, and briefly discussed in Section 3.1. We will identify the algebras for the subprobability functor through smooth equivalence relations and through positive convex structures in this chapter, first through the equivalence relations they induce on the set of subprobability measures. This will be a vehicle for an identification of these algebras without having to refer to the underlying probabilistic structure. It is done initially for the subprobability functor and, with some small adjustments, for the probability functor as well. We provide some examples to illustrate the algebras. Finally, the left adjoint of the forgetful functor that assigns each algebra the underlying Polish space is identified; it is just the functor that maps each Polish space to all its subprobabilities (with the monad’s multiplication as the associated algebra). We work in this chapter in the category cPol of Polish spaces with continuous maps as morphisms. A possible and desirable extension to the discussion here would be identification of Eilenberg-Moore algebras for the subprobability functor on analytic spaces with Borel measurable maps as morphisms.