Multiplicative structures on cohomology theories have been exploited fruitfully throughout the history of algebraic topology, ranging from the basic and classical argument that the Hopf maps are stably non-trivial to the more recent (and very much non-basic) use of equivariant power operations in the solution of the Kervaire invariant one problem due to Hill--Hopkins--Ravenel. The latter crucially relies on a refined notion of `genuine commutative algebras' in equivariant spectra, containing more information (in the form of `twisted power operations') than just E_\infty algebras in the \infty-category of spectra. While originally these objects were defined using well-behaved pointset models of spectra, in more recent years an alternative \infty-categorical approach has been suggested by Bachmann-Hoyois and Nardin-Shah. In this talk I will report on joint work with Sil Linskens and Phil Pützstück, in which we construct an equivalence between the two suggested definitions of `genuine commutative G-ring spectra.' I will further explain how our methods yield an \infty-categorical description of Schwede's `ultra-commutative global ring spectra' (again originally defined by a pointset model), and how this allows one to make precise the idea that an ultra-commutative global ring spectrum ought to be a compatible family of genuine commutative G-E_\infty ring spectra for all (finite) groups G.