Higher symmetries in abstract stable homotopy theories

Abstract

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.