A uniqueness theorem for stable homotopy theory

In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of the stable homotopy groups of spheres. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space QS^0 to the endomorphism space is a weak equivalence

On homotopy varieties

Given an algebraic theory \ct, a homotopy \ct-algebra is a simplicial set where all equations from \ct hold up to homotopy. All homotopy \ct-algebras form a homotopy variety. We give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, To\"{e}n and Vezzosi.Comment: Proposition 4.5 is not valid; see Remark 4.5(e) in the new version. All other results are correct but there are gaps in proofs. They are fixed by reducing simplicial categories to fibrant ones and replacing homotopy colimits by fibrant ones, as wel

Homotopy limits of model categories and more general homotopy theories

Generalizing a definition of homotopy fiber products of model categories, we give a definition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homotopy fiber products, we prove that such a homotopy limit does in fact correspond to the usual homotopy limit, when we work in a more general model for homotopy theories in which they can be regarded as objects of a model category.Comment: 10 pages; a few minor changes made. arXiv admin note: text overlap with arXiv:0811.317