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 in So{\l}tan's sense from a discrete quantum space to a partially ordered set is canonically equipped with quantum preorder in Weaver's sense. In particular, the quantum power set of a quantum set is so ordered. As an application, we show that each quantum poset embeds into its quantum power set.