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Fuzzy Presheaves are Quasitoposes

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Graph Transformation (ICGT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13961))

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Abstract

Quasitoposes encompass a wide range of structures, including various categories of graphs. They have proven to be a natural setting for reasoning about the metatheory of algebraic graph rewriting. In this paper we propose and motivate the notion of fuzzy presheaves, which generalises fuzzy sets and fuzzy graphs. We prove that fuzzy presheaves are rm-adhesive quasitoposes, proving our recent conjecture for fuzzy graphs. Furthermore, we show that simple fuzzy graph categories are quasitoposes.

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Notes

  1. 1.

    We moreover conjecture that PBPO\(^{+}\) subsumes SPO in quasitoposes [30, Rem. 26].

  2. 2.

    See e.g. [2, Def. 6.7], [45, Def. 24.1], or [24, 8.1].

  3. 3.

    Note that completeness and cocompleteness are equivalent for lattices [2, Def. 6.8].

  4. 4.

    The symbol \(\blacksquare \) denotes that the proof is in the extended version on arXiv [38]. A sketch of the proof is in [24, p.37-38].

  5. 5.

    Also known as quasiadhesive.

  6. 6.

    A graph homomorphism \(f : G \rightarrow H\) reflects edges if, for every \(v,w \in G(V)\), every edge between \(f_V(v)\) and \(f_V(w)\) has an f-preimage.

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Acknowledgments

We thank Helle Hvid Hansen for discussions and valuable suggestions. The authors received funding from the Netherlands Organization for Scientific Research (NWO) under the Innovational Research Incentives Scheme Vidi (project. No. VI.Vidi.192.004).

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  1. Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Aloïs Rosset, Roy Overbeek & Jörg Endrullis

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  1. Aloïs Rosset
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Correspondence to Aloïs Rosset .

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  1. King's College London, London, UK

    Maribel Fernández

  2. Singapore Management University, Singapore, Singapore

    Christopher M. Poskitt

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Rosset, A., Overbeek, R., Endrullis, J. (2023). Fuzzy Presheaves are Quasitoposes. In: Fernández, M., Poskitt, C.M. (eds) Graph Transformation. ICGT 2023. Lecture Notes in Computer Science, vol 13961. Springer, Cham. https://doi.org/10.1007/978-3-031-36709-0_6

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  • DOI: https://doi.org/10.1007/978-3-031-36709-0_6

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