Categorical (Co)Limits of Quantum Graphs
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914562624716800 |
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| author | Zhu, Jennifer |
| author_facet | Zhu, Jennifer |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13019 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Categorical (Co)Limits of Quantum Graphs Zhu, Jennifer Operator Algebras Category Theory 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. |
| title | Categorical (Co)Limits of Quantum Graphs |
| topic | Operator Algebras Category Theory |
| url | https://arxiv.org/abs/2605.13019 |