Boardman–Vogt
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| Boardman–Vogt | |
|---|---|
| Name | Boardman–Vogt |
| Introduced | 1960s |
| Authors | J. Michael Boardman; Rainer Vogt |
| Field | Algebraic topology; Category theory; Homotopy theory |
Boardman–Vogt is a construction in Algebraic topology and Category theory introduced by J. Michael Boardman and Rainer Vogt that organizes homotopy-coherent algebraic structures via operadic and categorical methods. The construction appears in work connected to Stable homotopy theory, Homotopy limit, Model category techniques and the study of Loop space machinery such as the May recognition principle and Stasheff polytope formulations. It provides a means to replace strict associative or strict monoid structures with homotopy-invariant replacements used across spectra, E∞-ring frameworks, and operadic resolutions.
Definition and Overview
The Boardman–Vogt construction produces a resolution of an operad or a topological category that yields homotopy-coherent versions of algebras for that operad, relating to the Bar construction and Cobar construction. In formal terms it associates to an operad P a new operad W(P) equipped with a natural augmentation W(P) → P which is a weak equivalence in appropriate model categories such as the topological or simplicial contexts. The process is closely tied to notions appearing in the work of Boardman and Vogt alongside subsequent developments by May, Kan, Quillen, and Hinich connecting to the Dwyer–Kan hammock localization and Drinfel'd-style homotopical algebra.
Historical Development
The construction originated in the late 1960s and early 1970s in papers by Boardman and Vogt addressing homotopy-invariant structures on loop and iterated loop objects, concurrent with work by Stasheff on A∞-spaces and May on operads. It was developed alongside the emergence of Model category theory by Quillen and influenced later formalizations by Dwyer, Kan, Berger, Fresse, and Markl. The Boardman–Vogt W-construction became a standard tool in treatments of E_n-structures, Deligne conjecture approaches by Kontsevich and Tamarkin, and was employed in categorical refinements by Lurie in higher category contexts and Toën in derived algebraic geometry settings.
Construction and Properties
The W-construction builds trees decorated by operations of an operad P with length parameters along edges, producing a topological (or simplicial) operad whose algebras model homotopy P-algebras. It respects homotopy equivalences under conditions analogous to those in Quillen frameworks and interacts with cofibrant and fibrant replacements studied by Hovey and Smith. Key properties include functoriality, an augmentation W(P)→P that is often a weak equivalence, preservation of cofibrancy in many enriched model structures, and compatibility with monoidal products as in work by Mandell, May, and Elmendorf. The construction admits comparisons to the Boardman homotopy invariant ideas in stable homotopy and to explicit combinatorial realizations such as Associahedron and Permutohedron-based decompositions studied by Stasheff and Tonks.
Applications and Examples
Boardman–Vogt resolutions are used to produce A∞- and E∞-structures on spaces and spectra in contexts considered by Adams, Hatcher, Ravenel, and Cohen. They appear in constructions of iterated loop space machines in the lineage of May and Segal and in homotopy-theoretic models of configuration spaces studied by Fulton, MacPherson, and Getzler. In derived contexts Boardman–Vogt techniques inform operadic resolutions used by Hinich, Keller, and Lurie for deformation theory and Hochschild cohomology calculations connected to the Deligne conjecture proofs by McClure and Smith. Concrete examples include W applied to the associative operad giving models for A∞-algebras and W applied to the commutative operad yielding models for homotopy-commutative structures used by Goerss and Hopkins.
Variants and Generalizations
Variants include simplicial, dendroidal, and enriched W-constructions adapted by Moerdijk, Weiss, Cisinski, and Caviglia. Generalizations appear in the form of homotopy-coherent nerve constructions studied by Cordier and Porter, as well as dendroidal adaptations by Moerdijk and Weiss linking to ∞-operad frameworks of Lurie and Toën. Enriched versions for symmetric monoidal categories are developed in the work of Fresse, Berger, and Muro to handle colored operads, props, and properads as in studies by Vallette and Merkulov.
Relationships to Other Operadic Structures
The Boardman–Vogt construction serves as a bridge between strict operads and homotopy-algebraic notions such as A∞-algebra, E_n-algebras, and homotopy-coherent categories studied by Kelly and Street. It compares to the Bar construction and Koszul duality frameworks developed by Ginzburg and Kapranov and interfaces with the Swiss-cheese operad investigations by Voronov and Kontsevich. The W-construction is often used in tandem with homotopical techniques from Dwyer and Kan and higher-categorical synthesis from Lurie to compare operadic homotopy types, model structures for symmetric monoidal contexts, and deformation theory applications in works by Hinich and Getzler.