A category of quantum posets

TL;DR

This paper introduces a new category of quantum posets that generalizes classical posets, demonstrating its mathematical properties and applications to quantum set theory and quantum power sets.

Contribution

It defines a category of quantum posets based on hereditarily atomic von Neumann algebras and explores its structural properties and applications.

Findings

The category is complete, cocomplete, and symmetric monoidal closed.

Quantum families of maps can be equipped with quantum preorders.

Quantum posets embed into their quantum power sets.

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.