Results in Mathematics

, Volume 72, Issue 1–2, pp 649–664 | Cite as

Betweenness Relations in a Categorical Setting

  • J. BrunoEmail author
  • A. McCluskey
  • P. Szeptycki


We apply a categorical lens to the study of betweenness relations by capturing them within a topological category, fibred in lattices, and study several subcategories of it. In particular, we show that its full subcategory of finite objects forms a Fraissé class implying the existence of a countable homogenous betweenness relation. We furthermore show that the subcategory of antisymmetric betweenness relations is reflective. As an application we recover the reflectivity of distributive complete lattices within complete lattices, and we end with some observations on the Dedekind–MacNeille completion.


Betweenness relations R-relations road systems antisymmetry separativity distributive closure Grothendieck firbration MacNeille completion lattices preorder partial order 

Mathematics Subject Classification



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The authors would like to extend their gratitude to Walter Tholen and Lili Shen for their many helpful suggestions to the paper. We would also like to extend our sincere gratitude to the referee for her/his very delicate review of our manuscript: we consider her/his comments as invaluable to our work and its exposition.


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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.University of BathBathUK
  2. 2.National University of IrelandGalwayIreland
  3. 3.York UniversityTorontoCanada

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