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 characterization 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.

We prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.