$t$-model structures

Abstract

For every stable model category M with a certain extra structure, we produce an associated model structure on the pro-category pro-M and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-M. Our motivating example is the category of pro-spectra.The extra structure referred to above is a t-model structure. This is a rigidification of the usual notion of a t-structure on a triangulated category. A t-model structure is a proper simplicial stable model category M with a t-structure on its homotopy category together with an additional factorization axiom.

A motivic version of the theorem of Fontaine and Wintenberger

“…Let be a category. As shown in [Fau06] any projectively cofibrant complex in is a retract of a complex that is the filtered colimit of bounded above complexes, each constituted by presheaves that are direct sums of representable ones.…”

Equivariant homotopy theory for pro–spectra

“…t -model categories are discussed in detail in [21]. They give rise to interesting model structures on pro-categories.…”

Model structures on pro-categories

“…The weak equivalences can be described in terms of pro-homotopy groups, but the reformulation is not quite as obvious as one might expect. See [9,17] for details.…”

“…A t-model category is a proper simplicial stable model category C with a tstructure on its (triangulated) homotopy category, together with a lift of the tstructure to C. This notion is studied in detail in [9], where it is shown that a particularly well-behaved filtered model structure on C (and thus a model structure on pro-C) can be associated to any t-model structure on C.…”

“…This article and [9] view n-equivalences and co-n-equivalences from a particular perspective involving model structures. We warn the reader that our definitions are different than those in [11], which also deals with model structures and a notion of n-equivalence.…”