  We define duality triples and duality pairs in compactly generated
triangulated categories and investigate their properties. This enables us to
give an elementary way to determine whether a class is closed under pure
subobjects, pure quotients and pure extensions, as well as providing a way to
show the existence of approximations. One key ingredient is a new
characterisation of phantom maps. We then introduce an axiomatic form of
Auslander-Gruson-Jensen duality, from which we define dual definable
categories, and show that these coincide with symmetric coproduct closed
duality pairs. This framework is ubiquitous, encompassing both algebraic
triangulated categories and stable homotopy theories. Accordingly, we provide
many applications in both settings, with a particular emphasis on silting
theory and stratified tensor-triangulated categories.
