ABSTRACT
This paper presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval.
The system $V_{\ast}(X)$ of Vietoris-Rips complexes for a data set $X$ is both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.
► BibTeX data
► References
[1] Michael Barr. Fuzzy set theory and topos theory. Canad. Math. Bull., 29 (4): 501–508, 1986. ISSN 0008-4395. 10.4153/CMB-1986-079-9.
https://doi.org/10.4153/CMB-1986-079-9
[2] Andrew J. Blumberg and Michael Lesnick. Universality of the homotopy interleaving distance. arXiv, 1705.01690, 2017. arXiv:1705.01690.
arXiv:1705.01690
[3] Gunnar Carlsson and Facundo Mémoli. Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res., 11: 1425–1470, 2010. ISSN 1532-4435.
[4] Didier Dubois, Walenty Ostasiewicz, and Henri Prade. Fuzzy sets: history and basic notions. In Fundamentals of fuzzy sets, volume 7 of Handb. Fuzzy Sets Ser., pages 21–124. Kluwer Acad. Publ., Boston, MA, 2000. 10.1007/978-1-4615-4429-6_2.
https://doi.org/10.1007/978-1-4615-4429-6_2
[5] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2009. ISBN 978-3-0346-0188-7. 10.1007/978-3-0346-0189-4. Reprint of the 1999 edition.
https://doi.org/10.1007/978-3-0346-0189-4
[6] John F. Jardine. Boolean localization, in practice. Documenta Mathematica, 1: 245–275, 1996. ISSN 1431-0635.
[7] John F. Jardine. Local persistence: homotopy theory of filtrations. Oberwolfach Reports, 5 (3): 1623–1625, 2008.
[8] John F. Jardine. Local Homotopy Theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 2015. 10.1007/978-1-4939-2300-7.
https://doi.org/10.1007/978-1-4939-2300-7
[9] Saunders Mac Lane and Ieke Moerdijk. Sheaves in geometry and logic. Universitext. Springer-Verlag, New York, 1994. ISBN 0-387-97710-4. 10.1007/978-1-4612-0927-0.
https://doi.org/10.1007/978-1-4612-0927-0
[10] Leland McInnes and John Healy. UMAP: uniform manifold approximation and projection for dimension reduction. 2018. arXiv:1802.03426.
arXiv:1802.03426
[11] David I. Spivak. Metric realization of fuzzy simplicial sets. Preprint, 2009.
Cited by
[1] King-Yin Yan, Lecture Notes in Computer Science 13154, 327 (2022) ISBN:978-3-030-93757-7.
The above citations are from Crossref's cited-by service (last updated successfully 2024-05-30 03:52:38). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch Crossref cited-by data during last attempt 2024-11-05 00:03:16: Could not fetch cited-by data for 10.32408/compositionality-1-3 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2024-11-05 00:03:16).
This Paper is published in Compositionality under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.