Bousfield localization

Bousfield localization
Name	Bousfield localization
Field	Algebraic topology
Introduced	1970s
Introduced by	A. K. Bousfield

Bousfield localization is a process in algebraic topology that modifies objects in a model category or homotopy category to force specified maps or homology theories to become equivalences, producing localized objects that reflect particular homological or homotopical information. It arose in the work of A. K. Bousfield and D. M. Kan during developments connected to the study of homology theory and the Eilenberg–Steenrod axioms, and it plays a central role alongside concepts from Quillen model category, Adams spectral sequence, Sullivan's nilpotent completion, and Morava K-theory.

Definition and motivation

The construction targets an object X in a model category such as the category of spectra or simplicial sets and produces a map X → LX satisfying a universal property: maps from LX detect equivalences with respect to a chosen homology theory E or a set S of maps. Motivations trace to problems studied by A. K. Bousfield, J. F. Adams, J. H. C. Whitehead, and G. W. Whitehead in attempts to isolate E-local information and to analyze localization phenomena in the spirit of Serre spectral sequence, Atiyah–Hirzebruch spectral sequence, and work on K-theory and cohomology operations. This formalism connects to methods used in the study of Postnikov towers, nilpotent spaces, and localization techniques employed by Sullivan, Quillen, and Bousfield–Kan.

Construction and existence

Existence proofs employ machinery from Quillen model categories, using left Bousfield localization of a model structure with respect to a set of maps S, producing a new model category with the same cofibrations and more weak equivalences. Foundational results involve work of A. K. Bousfield, D. M. Kan, and later developments by H. H. Baues, C. Rezk, and M. Hovey, linking to the concept of cellularization used by J. P. May and M. Cole. Alternate constructions use homotopy colimits, transfinite compositions, and localization functors characterized by universal properties analogous to localization in commutative algebra though executed in the homotopical context studied by Grothendieck and Serre. Criteria for existence often appeal to set-theoretic conditions related to the presence of generating cofibrations and to smallness conditions developed by Hovey and by J. Smith in the context of combinatorial model categories and accessible categories influenced by Grothendieck's ideas.

Examples and special cases

Important examples include localization with respect to ordinary homology H_*(-; A) for a ring A, producing H_*(-; A)-local objects studied by A. K. Bousfield, and localization with respect to Morava K-theory K(n) or Johnson–Wilson theory E(n), central to the chromatic homotopy theory program of J. Morava, J. Frank Adams, and N. Ravenel. p-localization and site-specific completions relate to work by Sullivan on localizations of nilpotent spaces and to the Rational homotopy theory of D. Sullivan and D. Quillen. In stable homotopy categories, localizations at homology theories connect to the Adams–Novikov spectral sequence, the Nilpotence theorem of D. C. Ravenel and E. S. Devinatz, and to telescopic localization considered by M. Hopkins and J. H. Smith.

Properties and functoriality

Localization functors L are idempotent and universal among maps that are equivalences for the chosen detecting data; these properties were formalized by A. K. Bousfield and J. H. C. Whitehead and are analogous to localization in the sense of C. Gabriel and M. Zisman’s localization of categories. Functoriality interacts with change-of-ring and change-of-spectrum phenomena studied by H. Miller, P. S. Goerss, and L. F. Goerss, with compatibility constraints expressed in terms of homotopy limits and colimits reminiscent of constructions used by P. May, M. Mandell, and B. Shipley. Further structural results connect localizations to exact triangles and Bousfield lattices appearing in work by P. Balmer and P. Strickland, and to smashing localizations and telescope conjectures investigated by M. Hopkins and N. Kuhn.

Applications in homotopy theory

Bousfield localization is applied to compute localized homotopy groups, to simplify computational problems in the Adams spectral sequence and Adams–Novikov spectral sequence, and to isolate chromatic layers in programs by M. Hopkins, E. S. Devinatz, and N. Strickland. It underlies approaches to the classification of thick subcategories in stable homotopy categories as pursued by T. Hopkins and J. Smith, and it appears in the study of module spectra over structured ring spectra such as A∞ and E∞ algebras developed by M. Mandell and J. P. May. Further applications include analyses of localization towers in the spirit of Postnikov and Whitehead, connections with localized model structures used by D. Dugger and B. Shipley, and uses in computations around elliptic cohomology studied by M. Ando and M. Hopkins.

Variants and generalizations

Variants include right Bousfield localization (cellularization) and localization with respect to classes of morphisms rather than single homology theories, developed in frameworks by H. H. Baues, C. Rezk, and J. Chorny, and extended to combinatorial model categories by M. Hovey and D. Dugger. Generalizations reach into higher category theory via localization of ∞-categories and presentable ∞-categories as treated in the work of J. Lurie, and they connect to derived localization in derived algebraic geometry studied by P. Deligne, A. Grothendieck, and B. Toen. Interactions with tensor-triangular geometry and Balmer spectra further broaden the landscape, linking to research by P. Balmer, M. Saorin, and B. Keller.

Category:Algebraic topology