×

A guide to tensor-triangular classification. (English) Zbl 1476.55042

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 145-162 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: Stable homotopy theory shines across pure mathematics, from topology to analysis, from algebra to geometry. This chapter discusses equivariant stable homotopy theory and Kasparov’s equivariant \(KK\)-theory. The original idea of classifying objects up to the ambient structure was born in topology, around Ravenel’s conjectures and the ‘chromatic’ theorems of Devinatz-Hopkins-Smith. The tt-classification was born in topology, more precisely in chromatic homotopy theory. A snapshot of tensor-triangular geometry as of the year 2010 can be found in [P. Balmer, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 85–112 (2011; Zbl 1235.18012)]. The comparison map was generalized in two directions. First by Dell’Ambrogio-Stevenson, by allowing grading by a collection of invertible objects instead of a single one. Secondly, higher comparison maps were defined by Sanders in order to refine the analysis of the fibers of ‘lower’ comparison maps, through an inductive process. A great deal of progress followed from the development of the idea of separable extensions of tt-categories.
For the entire collection see [Zbl 1468.55001].

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G80 Derived categories, triangulated categories
55P42 Stable homotopy theory, spectra
18F99 Categories in geometry and topology
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
PDFBibTeX XMLCite
Full Text: DOI arXiv