Lifting homotopy T-algebra maps to strict maps

Abstract

The settings for homotopical algebra—categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ 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 π0 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 E2-term to be calculable in terms of Quillen cohomology groups.We provide worked examples in G-spaces, G-spectra, rational E∞ algebras, and A∞ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of E∞ ring spectra to the category of H∞ 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∞(Σ+∞CokerJ,LK(2)R).