2023

Fuzzy Presheaves Are Quasitoposes
RossetAloı̈s, Roy Overbeek, and Jörg Endrullis
In: Graph Transformation - 16th International Conference, ICGT 2023, Held as Part of STAF 2023, Leicester, UK, July 19-20, 2023, Proceedings Fernández, Maribel and Poskitt, Christopher M. (eds.) , pp. 102–122 , Springer (2023)
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Summary

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 graphs categories are quasitoposes.
Bibtex
```
@inproceedings{graph:rewriting:fuzzy:presheaves:2023,
  author = {Rosset, Alo{\"{\i}}s and Overbeek, Roy and Endrullis, J{\"{o}}rg},
  editor = {Fern{\'{a}}ndez, Maribel and Poskitt, Christopher M.},
  title = {Fuzzy Presheaves are Quasitoposes},
  booktitle = {Graph Transformation - 16th International Conference, {ICGT} 2023,
                    Held as Part of {STAF} 2023, Leicester, UK, July 19-20, 2023, Proceedings},
  series = {Lecture Notes in Computer Science},
  volume = {13961},
  pages = {102--122},
  publisher = {Springer},
  year = {2023},
  url = {https://doi.org/10.1007/978-3-031-36709-0\_6},
  doi = {10.1007/978-3-031-36709-0\_6},
  biburl = {https://dblp.org/rec/conf/gg/RossetOE23.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org},
  keywords = {rewriting, category theory},
  type = {conference}
}
```
Digital Object Identifier
```
10.1007/978-3-031-36709-0_6
```

Characterisation of Lawvere-Tierney Topologies on Simplicial Sets, Bicolored Graphs, and Fuzzy Sets

RossetAloı̈s, Helle Hvid Hansen, and Jörg Endrullis

In: Logic, Language, and Computation - 14th International Tbilisi Symposium, TbiLLC 2023, Telavi, Georgia, September 18-22, 2023, Revised Selected Papers Hogeweg, Lotte, Jalali, Raheleh, Sedlár, Igor, and Schwarz, Lena (eds.) , pp. 258–284 , Springer (2023)

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Summary

Simplicial sets generalise many categories of graphs. In this paper, we give a complete characterisation of the Lawvere-Tierney topologies on (semi-)simplicial sets, on bicolored graphs, and on fuzzy sets. We apply our results to establish that 'partially simple' simplicial sets and 'partially simple' graphs form quasitoposes.

Bibtex

@inproceedings{topos:lawvere:tierney:2023,
  author = {Rosset, Alo{\"{\i}}s and Hansen, Helle Hvid and Endrullis, J{\"{o}}rg},
  editor = {Hogeweg, Lotte and Jalali, Raheleh and Sedl{\'{a}}r, Igor and Schwarz, Lena},
  title = {Characterisation of Lawvere-Tierney Topologies on Simplicial Sets,
                    Bicolored Graphs, and Fuzzy Sets},
  booktitle = {Logic, Language, and Computation - 14th International Tbilisi Symposium,
                    TbiLLC 2023, Telavi, Georgia, September 18-22, 2023, Revised Selected
                    Papers},
  series = {Lecture Notes in Computer Science},
  volume = {15661},
  pages = {258--284},
  publisher = {Springer},
  year = {2023},
  url = {https://doi.org/10.1007/978-3-032-14339-6\_14},
  doi = {10.1007/978-3-032-14339-6\_14},
  biburl = {https://dblp.org/rec/conf/tbillc/RossetHE23.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org},
  keywords = {category theory},
  type = {conference}
}

Digital Object Identifier

10.1007/978-3-032-14339-6_14