Results Math 72 (2017), 649–664
2017 Springer International Publishing
published online March 23, 2017
Results in Mathematics
Betweenness Relations in a Categorical
, A. McCluskey, and P. Szeptycki
Abstract. We apply a categorical lens to the study of betweenness rela-
tions 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 Fraiss´e 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 lat-
tices within complete lattices, and we end with some observations on the
Mathematics Subject Classiﬁcation. 18D35.
Keywords. Betweenness relations, R-relations, road systems, antisymme-
try, separativity, distributive closure, Grothendieck ﬁrbration, MacNeille
completion, lattices, preorder, partial order.
1. Introduction and Background
The study of betweenness relations dates back as far the late 1800’s  with
sporadic revivals throughout the last century whose focus lie in characterising
certain partial orders in terms of the betweenness relations they generate (see
for example [5,7,9,10,15,17,18]). An excellent modern approach can be found
in  where Bankston explores a large variety of settings in which betweenness
relations arise, with emphasis on ordered sets, metric spaces and continuum
topology. The author makes a strong case for considering the simple notion
of a road system on an arbitrary set as a natural approach to generating the
intuitive idea of betweenness relations as ternary relations. In the broadest of
interpretations, for a point b to lie between points a and c it must be that
any way to get from a to c must inevitably go though b. Intuitively, given an
arbitrary point, it is desirable that no other point lies between it and itself.