[Paper Review] An introduction to Bousfield localization

Tyler Lawson|arXiv (Cornell University)|Feb 10, 2020

Homotopy and Cohomology in Algebraic Topology16 references4 citations

TL;DR

This paper provides a comprehensive introduction to Bousfield localization in homotopy theory, explaining its categorical foundations, construction via the small object argument, and applications in algebraic topology. It establishes that Bousfield localization preserves monoidal structures under mild conditions, enabling the construction of localized ∞-categories with compatible monoidal operations.

ABSTRACT

In this article we discuss Bousfield localization, beginning with definitions in terms of mapping spaces and working up to a discussion of how they can be constructed when we have access to the small object argument. We also discuss Bousfield localization in the presence of multiplicative structure. Our goal is to place an emphasis on examples of various types. This is an expository article, written to be part of an upcoming book.

Motivation & Objective

• To provide a self-contained introduction to Bousfield localization in homotopy theory, emphasizing its role in simplifying and organizing algebraic topology.
• To explain how localization functors can be constructed using the small object argument, particularly in model categories and ∞-categories.
• To demonstrate the compatibility of Bousfield localization with monoidal structures, especially in ∞-categories equipped with an ∞-operad.
• To illustrate the utility of localization in decomposing spaces and spectra via rationalization, p-localization, and completion, and in recovering global information from local data.
• To clarify the universal properties of localization functors and their role in simplifying homotopical computations through fibrant replacement and universal mapping spaces.

Proposed method

• Use categorical localization as a foundation, defining localization via universal property of inverting a class of maps.
• Apply the small object argument to construct left Bousfield localizations in model categories, ensuring existence of localization functors.
• Work within ∞-categories enriched in spaces, using mapping spaces and homotopy categories to define and analyze local objects.
• Leverage Lurie’s theory of ∞-operads and ∞-monoidal categories to formalize compatibility of localization with monoidal structures.
• Use the universal property of fibrant replacement in Quillen’s model structure to show that rationalization and p-localization arise as fibrant replacements.
• Apply the arithmetic fracture square to reconstruct a space from its rational and p-adic localizations, using homotopy pullbacks.

Experimental results

Research questions

• RQ1 How can Bousfield localization be systematically constructed in model categories and ∞-categories using the small object argument?
• RQ2 Under what conditions does Bousfield localization preserve monoidal structures in ∞-categories?
• RQ3 How does the compatibility of localization with products and loop spaces enable fiberwise localization in categories of spaces over a base?
• RQ4 What is the role of the small object argument in ensuring the existence of localization functors in homotopical contexts?
• RQ5 How do arithmetic fracture techniques allow the reconstruction of global homotopy types from their rational and p-adic localizations?

Key findings

• Bousfield localization can be constructed in any model category via the small object argument, ensuring the existence of localization functors that universally invert a given class of maps.
• The subcategory of local objects in a Bousfield localization inherits a natural ∞-monoidal structure when the localization preserves $L$-equivalences in each variable, as formalized in Theorem 12.22.
• For any localization $L$ on spaces, the product $LX imes LY$ is equivalent to $L(X imes Y)$, showing that the cartesian product is compatible with all localization functors.
• Fiberwise localization on spaces over a grouplike $E_n$-space $B$ is compatible with the Day convolution monoidal structure, making it an $E_n$-monoidal functor.
• The arithmetic fracture square allows a simply-connected space $X$ to be recovered as a homotopy pullback of its rationalization $X_{\mathbb{Q}}$ and $p$-completions $X^\wedge_p$, demonstrating the global recovery of homotopy types from local data.
• Quillen’s model structure on spaces, where weak equivalences are isomorphisms on rational homology, realizes rationalization as a fibrant replacement, highlighting the universality of localization functors.