A category of quantum posets

Abstract

We investigate a category of quantum posets that generalizes the category of posets and monotone functions. Up to equivalence, its objects are hereditarily atomic von Neumann algebras equipped with quantum partial orders in Weaver’s sense. We show that this category is complete, cocomplete and symmetric monoidal closed. As a consequence, any discrete quantum family of maps from a discrete quantum space to a partially ordered set is canonically equipped with a quantum preorder. In particular, the quantum power set of a quantum set is canonically a quantum poset. We show that each quantum poset embeds into its quantum power set in complete analogy with the classical case.