[Paper Review] On the Dreaded Right Bousfield Localization

Clark Barwick|ArXiv.org|Aug 25, 2007

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper establishes the existence of right Bousfield localizations in right semimodel categories, removing the need for right properness in model categories. It proves that such localizations exist without properness assumptions and applies the result to construct a model for the homotopy limit of left Quillen presheaves as a right semimodel category.

ABSTRACT

I verify the existence of right Bousfield localizations of right semimodel categories, and I apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right semimodel category.

Motivation & Objective

• To resolve the limitation that right Bousfield localizations typically require right properness in model categories.
• To extend the theory of right Bousfield localization beyond proper model categories, particularly in contexts where right properness fails.
• To provide a framework for constructing homotopy limits of left Quillen presheaves using right semimodel categories.
• To generalize the classical right Bousfield localization to settings where full model category structures are not available.
• To establish a foundation for homotopy-theoretic constructions in non-proper contexts via semimodel categories.

Proposed method

• Introduces the concept of a right semimodel category, where factorization and lifting axioms hold only when the target is fibrant.
• Applies the general theory of structured homotopical categories to define right Bousfield localization in this context.
• Constructs the localization as a right semimodel category by verifying the necessary axioms without assuming right properness.
• Uses cofibrant homotopy generators and essential images under derived functors to build a small set of generators for the localized category.
• Applies the localization to left Quillen presheaves by showing that homotopy cartesian sections can be built from finite homotopy colimits of generators.
• Establishes that the homotopy limit of a left Quillen presheaf admits a model structure as a right semimodel category via this construction.

Experimental results

Research questions

• RQ1 Can right Bousfield localizations be constructed in the absence of right properness in model categories?
• RQ2 Is there a general framework for right Bousfield localization that applies to non-right-proper categories?
• RQ3 Can the homotopy limit of a left Quillen presheaf be modeled using a right semimodel category structure?
• RQ4 What conditions ensure that a right semimodel category supports a well-behaved localization with respect to a set of objects?
• RQ5 How can cofibrant generators be used to build a small set of representatives for the homotopy limit of a presheaf?

Key findings

• Right Bousfield localizations exist as right semimodel categories even when the original category is not right proper.
• The localization is constructed by verifying all model category axioms except for factorization and lifting when the target is not fibrant.
• The homotopy limit of a left Quillen presheaf admits a model structure as a right semimodel category via this localization procedure.
• Any homotopy cartesian section of the presheaf can be expressed as a homotopy colimit of sections built from cofibrant-fibrant representatives of essential images.
• A small set $ G $ of cofibrant homotopy cartesian sections generates the homotopy category of the homotopy limit.
• The construction is independent of the choice of cofibrant generators and depends only on the essential images of the functors in the presheaf.