Homotopical resolutions associated to deformable adjunctions
Andrew J. Blumberg, Emily Riehl
TL;DR
This paper introduces homotopical resolutions linked to deformable adjunctions, providing new models for completion and cocompletion that enhance spectral sequence computations and Goodwillie calculus in model categories.
Contribution
It develops a novel derived bar and cobar construction for adjunctions, extending homotopical models of monads and comonads in the context of weak equivalences.
Findings
New homotopical models for completion and cocompletion
Applications to spectral sequences in derived contexts
Enhanced tools for Goodwillie calculus in model categories
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.
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