Bousfield Localization

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.

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Nondurable K-Theory Equivalence and Bousfield Localization*

K-Theory ◽

10.1023/a:1014083003668 ◽

2001 ◽

Vol 24 (4) ◽

pp. 397-410 ◽

Cited By ~ 3

Author(s):

Lisa Langsetmo ◽

Don Stanley

Keyword(s):

K Theory ◽

Bousfield Localization

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Locally constant functors

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410900262x ◽

2009 ◽

Vol 147 (3) ◽

pp. 593-614 ◽

Cited By ~ 5

Author(s):

DENIS–CHARLES CISINSKI

Keyword(s):

Homotopy Theory ◽

Model Category ◽

Model Categories ◽

Combinatorial Model ◽

Constant Coefficients ◽

Kan Extensions ◽

Bousfield Localization

AbstractWe study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

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