A category of quantum posets
Andre Kornell, Bert Lindenhovius, Michael Mislove
TL;DR
The paper defines a category of quantum posets ($\mathbf{qPOS}$) by combining hereditarily atomic von Neumann algebras with Weaver's quantum relations, and develops its categorical structure. It proves that $\mathbf{qPOS}$ is complete, cocomplete, and symmetric monoidal closed, enabling canonical constructions of limits, colimits, tensor products, and internal homs. A key outcome is that every discrete quantum family of maps carries a natural quantum preorder, and the quantum power set of a quantum set becomes a quantum poset with a canonical order, into which any quantum poset embeds. The results provide a robust denotational-semantics framework for quantum programming languages and offer a foundation for quantum topology via a quantum power-set construction. Overall, the work advances noncommutative generalizations of posets, function spaces, and power sets with strong categorical properties.
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 Sołtan'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.