A-local symmetric spectra

This paper is about importing the stable homotopy theory of symmetric spectra [4] and more generally presheaves of symmetric spectra [8] into the Morel-Voevodsky stable category [9], [11], [12]. Loosely speaking, the latter is the result of formally inverting the functor X 7→ T∧X on the category of pointed simplicial presheaves on the smooth Nisnevich site of a field within the Morel-Voevodsky A-local homotopy theory, where T is defined to be the quotient of schemes A/(A − 0). The MorelVoevodsky stable homotopy theory is exotic in at least two ways: it lives within a localized homotopy theory of simplicial presheaves, and the object T is not a circle in any sense, but is rather weakly equivalent within the A-local theory to an honest suspension S ∧Gm of the scheme underlying the multiplicative group. Smashing with T is thus a combination of topological and geometric suspensions. The Morel-Voevodsky stable category is fundamental for Voevodsky’s proof of the Milnor Conjecture [11]. It arises from a suitable notion of stable equivalence, subsumed by a proper closed simplicial model structure on the category of presheaves of T -spectra on a smooth Nisnevich site. A presheaf of T -spectra X consists of pointed simplicial presheaves X, n ≥ 0, together with bonding maps T ∧Xn → X. A symmetric object in this category, or rather a presheaf of symmetric T -spectra, is a presheaf of T -spectra Y , equipped with symmetric group actions Σn × Y n → Y n in all levels such that all composite bonding maps T∧p ∧ X → X are (Σp×Σn)-equivariant. The main new results of this paper assert that the category of presheaves of symmetric T -spectra carries a notion of stable equivalence within the A-local theory which is part of a proper closed simplicial model structure (Theorem 4.18), and such that the forgetful functor to presheaves of T -spectra induces an equivalence of the stable homotopy category for presheaves of symmetric T -spectra with the Morel-Voevodsky stable category (Theorem 5.14). This collection of results gives a category which models the Morel-Voevodsky stable category, and also has a symmetric monoidal smash product.