Homotopical resolutions associated to deformable adjunctions

Abstract

Given an adjunction connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calculus in general model categories.

22 pages; v2 is the final journal version, with expository improvements suggested by the referee

References in corpus (4)

• Higher Topos Theory
• A general context for Goodwillie Calculus
• Cofibrance and Completion
• A classification of Taylor towers of functors of spaces and spectra

Cited by in corpus (14)

• Algebraic weak factorisation systems I: accessible AWFS
• Quillen adjunctions induce adjunctions of quasicategories
• Lifting homotopy T-algebra maps to strict maps
• A model structure on internal categories in simplicial sets
• Homotopical algebra for Lie algebroids
• Derived Koszul duality and TQ-homology completion of structured ring spectra
• Suspension spectra and higher stabilization
• Iterated delooping and desuspension of structured ring spectra
• Operads, modules and topological field theories
• Iterated suspension spaces and higher Freudenthal suspension
• Derived Commutator Complete Algebras and Relative Koszul Duality for Operads
• Bousfield localisation and colocalisation of one-dimensional model structures
• The partition poset complex and the Goodwillie derivatives of the identity in spaces
• $\infty$-categorical monadicity and descent