This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as ΔopPshv(Sm)𝔸1 and Δop
PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM
eff
nis
≤0 of DM
eff
nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ-closed classes, and develops their basic properties. The theory of ̅Δ-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.
Bousfield Localization and Algebras over Colored Operads
