Lifting homotopy T-algebra maps to strict maps

• Published in: arXiv.org 2014-07
• Main Authors: Johnson, Niles, Noel, Justin
• Format: Article
• Language: English
• Subjects: Algebra; Homology; Homomorphisms; Homotopy theory; Hypotheses; Rings (mathematics); Spectra
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1301.1511

The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, \(A_\infty\) spaces, \(E_\infty\) ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple \(T\). In such cases, \(T\) is acting on a nice simplicial model category in such a way that \(T\) descends to a monad on the homotopy category and defines a category of homotopy \(T\)-algebras. In this setting there is a forgetful functor from the homotopy category of \(T\)-algebras to the category of homotopy \(T\)-algebras. Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy \(T\)-algebra map to a strict map of \(T\)-algebras. Once we have a map of \(T\)-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of \(T\)-algebra maps and the edge homomorphism on \(\pi_0\) is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the \(E_2\)-term to be calculable in terms of Quillen cohomology groups. We provide worked examples in \(G\)-spaces, \(G\)-spectra, rational \(E_\infty\) algebras, and \(A_\infty\) algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of \(E_\infty\) ring spectra to the category of \(H_\infty\) ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space \(E_\infty(\Sigma^\infty_+ \mathrm{Coker}\, J, L_{K(2)} R)\).