Categorical (Co)Limits of Quantum Graphs

TL;DR

This paper characterizes quantum graphs using operator space theory and introduces a categorical framework for their (co)limits, providing a representation-free approach to quantum graph morphisms.

Contribution

It offers a new, representation-independent characterization of quantum graphs and their morphisms, and develops a categorical theory of (co)limits for quantum graphs.

Findings

Quantum graphs are characterized as left ideals in an extended Haagerup tensor product.

A representation-free characterization of quantum graph morphisms is established.

A categorical framework for (co)limits of quantum graphs is introduced.

Abstract

We begin with the characterization of quantum graphs as left ideals in $\mathcal M \otimes_{eh} \mathcal M$ (the extended Haagerup tensor product of $\mathcal M$ with itself) to avoid technicalities surrounding representation dependence of quantum graphs. These left ideals roughly correspond to a canonical complement of a quantum graph. Using these left ideals and some operator space theory, we find a new, representation-free characterization of a morphism of quantum graphs compatible with previous representation-dependent morphisms. A notion of categorical (co)limit of quantum graphs follows. We also briefly explore an alternative quantization of graphs as bimodules over $C^*$-algebras ($C^*$-graphs), mostly to emphasize the point that a morphism of $C^*$-graphs is not a morphism of $C^*$-correspondences.