Stochastic order on metric spaces and the ordered Kantorovich monad
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In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. Our proof of antisymmetry of the stochastic order on these spaces is more general than previously known results in this direction. The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances. We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone affine maps. Considering the category of L-ordered metric spaces as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively. In the unordered case, we had identified the Wasserstein space as the colimit of the spaces of empirical distributions of finite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.
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