Abstract
This dissertation explores how universality in category theory can be used to understand structural aspects
of graphs. We use four equivalent presentations of universality (universal morphisms, representability,
limits and colimits, and adjunction) to describe connections between various subcategories of Rel, the
category of internal relations on sets and DGrph, the category of internal graphs on sets. With the view
that relations are specialised graphs, we present relations structurally as pairs of jointly monic maps, and
graphs as pairs of maps with the joint monicity condition relaxed. Known adjoints to inclusion functors
between various categories of relations are examined. We prove that Rel is a (full) reflective subcategory of
DGrph, justifying the study of reflections and coreflections in relations, as a mechanism for shedding light
on possible reflections and coreflections that exist among certain subcategories of DGrph. A summary of
the reflections and coreflections studied is presented in Figure 11. The dissertation concludes with further
research proposals presented.