Higher symmetries in abstract stable homotopy theories. (English) Zbl 1469.55015
Šťovíček, Jan (ed.) et al., Representation theory and beyond. Workshop and 18th international conference on representations of algebras, ICRA 2018, Prague, Czech Republic, August 13–17, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 758, 91-193 (2020).
Summary: 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.
For the entire collection see [Zbl 1461.16004].
For the entire collection see [Zbl 1461.16004].
MSC:
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
| 18G80 | Derived categories, triangulated categories |
| 55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |
| 18-02 | Research exposition (monographs, survey articles) pertaining to category theory |
| 55P42 | Stable homotopy theory, spectra |
| 55U40 | Topological categories, foundations of homotopy theory |
| 16E35 | Derived categories and associative algebras |
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