  This survey offers an overview of an on-going project on uniform symmetries
in abstract stable homotopy theories. This project has calculational,
foundational, and representation-theoretic aspects, and key features of this
emerging field on abstract representation theory include the following. First,
generalizing the classical focus on representations over fields, it is
concerned with the study of representations over rings, differential-graded
algebras, ring spectra, and in more general abstract stable homotopy theories.
Second, restricting attention to specific shapes, it offers an explanation of
the axioms of triangulated categories, higher triangulations, and monoidal
triangulations. This has led to fairly general results concerning additivity of
traces. Third, along similar lines of thought it suggests the development of
abstract cubical homotopy theory as an additional calculational toolkit. An
interesting symmetry in this case is given by a global form of Serre duality.
Fourth, abstract tilting equivalences give rise to non-trivial elements in
spectral Picard groupoids and hence contribute to their calculation. And,
finally, it stimulates a deeper digression of the notion of stability itself,
leading to various characterizations and relative versions of stability.
