A category of quantum posets
arXiv:2101.11184 · doi:10.1016/j.indag.2022.07.001
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.
34 pages; added quantum power set functor, extended introduction
References in corpus (16)
- A Theory of Quantum Error-Correcting Codes
- Quantum Symmetry Groups of Finite Spaces
- Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function
- Estimating quantum chromatic numbers
- Graph Homomorphisms for Quantum Players
- Quantum families of maps and quantum semigroups on finite quantum spaces
- A compositional approach to quantum functions
- Quantum source-channel coding and non-commutative graph theory
- The quantum-to-classical graph homomorphism game
- Quantum sets
- On monoids in the category of sets and relations
- A von Neumann algebra approach to quantum metrics
- Random quantum graphs
- Deciding the existence of perfect entangled strategies for nonlocal games
- Quantum CPOs
- Hereditarily antisymmetric operator algebras