Probabilistic Metric Spaces as enriched categories

In this paper we investigate Cauchy completeness and exponentiablity for quantale enriched categories, paying particular attention to probabilistic metric spaces.

💡 Research Summary

The paper investigates Cauchy completeness and exponentiability in categories enriched over a quantale, with a special focus on probabilistic metric spaces. After recalling the definition of a quantale (a complete lattice equipped with a commutative, associative tensor ⊗ having a neutral element k that distributes over arbitrary suprema) and presenting basic examples (the two‑element Boolean algebra, the extended non‑negative reals with addition, the unit interval with multiplication), the authors introduce V‑categories. A V‑category X=(X,a) consists of a set X together with a “distance” map a:X×X→V satisfying k≤a(x,x) and a(x,y)⊗a(y,z)≤a(x,z). V‑functors are maps preserving this structure, and V‑modules (also called distributors or profunctors) are bimodules ϕ:X⇸Y satisfying left‑ and right‑action inequalities. The paper emphasizes that a V‑module ϕ can be viewed as a V‑functor Xᵒᵖ⊗Y→V, and that the Yoneda embedding y_X:X→