Abstract: T-spectra, or spectrum objects with generalized "suspension" parameters, first appeared in the construction of motivic stable categories. The motivic stable model structure lives within a localized model structure of simplicial presheaves in which the affine line is formally collapsed to a point, and the suspension object is the projective line. Because of these constraints and limitations of the tools then at hand, the original construction of the motivic stable category was technical, and made heavy use of the Nisnevich descent theorem.

This talk will begin with a general introduction to the concepts around T-spectra. I shall display a short list of axioms on the parameter object T and the ambient f-local model category which together lead to the construction of a well behaved f-local stable model structure of T-spectra. Examples include the motivic stable category, but the construction is much more general. The resulting stable category has many of the basic calculational features of the motivic stable category, including slice filtrations. The overall construction method is to suitably localize an easily defined strict model structure for T-spectra. The localization trick is an old idea of Jeff Smith, but assumptions (the axioms) are required for the recovery of normal features of stable homotopy theory from the localized structure.