Spectra and T-spectra

Abstract

Presheaves of spectra have been in use for some time. The original applications were in algebraic K -theory, but they now appear in various other areas. A simple existence proof for the stable model structure for presheaves of spectra appears in the first section of this chapter.

Presheaves of spectra are defined by the classical parameter object, namely the circle. The need for more general parameter objects T, such as the projective line, first appeared in motivic cohomology theory, which setting is also localized in the sense that the affine line is collapsed to a point.

This chapter displays a general approach to constructing stable model structures for categories of T -spectra, in localized settings. The methods of Chapter 7 construct the ambient stable model structures, but interpretation of these structures requires assumptions on both the parameter object and the underlying localization for simplicial presheaves.

This general technique applies to T -spectrum objects in abelian settings, and in particular to T -spectrum objects in presheaves with transfers. The resulting stable category is Voevodsky's big category of motives.