Characterisation of Lawvere-Tierney Topologies on Simplicial Sets, Bicolored Graphs, and Fuzzy Sets
Alo\"is Rosset, Helle Hvid Hansen, J\"org Endrullis
TL;DR
This paper characterizes Lawvere-Tierney topologies across various structures like simplicial sets, bicolored graphs, and fuzzy sets, revealing their categorical properties and establishing quasitopos structures.
Contribution
It provides a complete characterization of Lawvere-Tierney topologies on these structures and shows that certain 'partially simple' objects form quasitoposes.
Findings
Complete characterization of Lawvere-Tierney topologies
Identification of quasitopos structures in 'partially simple' objects
Application of results to categorical properties of graphs and fuzzy sets
Abstract
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.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Rough Sets and Fuzzy Logic · Fuzzy and Soft Set Theory