We survey work in category theory and coalgebra on how to extend a functor from maps to relations. This relation lifting has a universal property, which is presented in some detail and guides us to generalisations to monotone and many-valued relations. As applications, it is shown how different notions of bisimulation, simulation and modal logics do arise.
['Alexander Kurz', 'Jiří Velebil']
Relation lifting, a survey