Fuzzy sets and presheaves
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866914901088272384 |
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| author | Jardine, J. F. |
| author_facet | Jardine, J. F. |
| contents | This note 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. A presheaf-theoretic method is used to show that the category of fuzzy sets is complete and co-complete, and to present explicit descriptions of classical fuzzy sets that arise as limits and colimits. The Boolean localization construction for sheaves and presheaves on a locale L specializes to a theory of stalks if L approximates the structure of a closed interval in the real line. The system V(X) of Vietoris-Rips complexes for a data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed in this paper, in stages. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_10314 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Fuzzy sets and presheaves Jardine, J. F. Category Theory This note 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. A presheaf-theoretic method is used to show that the category of fuzzy sets is complete and co-complete, and to present explicit descriptions of classical fuzzy sets that arise as limits and colimits. The Boolean localization construction for sheaves and presheaves on a locale L specializes to a theory of stalks if L approximates the structure of a closed interval in the real line. The system V(X) of Vietoris-Rips complexes for a data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed in this paper, in stages. |
| title | Fuzzy sets and presheaves |
| topic | Category Theory |
| url | https://arxiv.org/abs/1904.10314 |