ABSTRACT

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. 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.