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

Aloïs Rosset, Helle Hvid Hansen, Jörg Endrullis

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.

Characterisation of Lawvere-Tierney Topologies on Simplicial Sets, Bicolored 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.
Paper Structure (23 sections, 10 theorems, 51 equations, 2 tables)

This paper contains 23 sections, 10 theorems, 51 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theory
  4. Category theory
  5. Topologies
  6. Toplogies on Simplicial Sets
  7. Simplicial Sets
  8. Topologies on n-dimensional semi-sSet
  9. Topologies on n-dimensional sSet
  10. Topologies on semi-sSet and sSet
  11. Separated elements and sheaves
  12. Topologies on bicolored graphs
  13. Topologies on Fuzzy sets
  14. Conclusion
  15. Related Work
  16. ...and 8 more sections

Key Result

lemma 1

In a topos $\mathsf{E}$, there is a bijection between LT-topologies and closure operators. Given an LT-topology $j$ and $A' \rightarrowtail A$, the closure $\overline{A'} \rightarrowtail A$ is defined by $\chi^{\overline{A'}} \vcentcolon= j \circ \chi^{A'}$. Given a closure operator $\tau$, the corr

Theorems & Definitions (60)

  • definition 1
  • definition 2
  • definition 3
  • definition 4: Awodey_2006
  • definition 5: Wyler_1991
  • definition 6
  • definition 7
  • definition 8
  • remark 1
  • definition 9: McLarty_1992_Elementary_categories_elementary_toposes
  • ...and 50 more