Zusammenfassung
The settings for homotopical algebra-categories such as simplicial groups, simplicial rings, A(infinity) spaces, E-infinity 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 ...
Zusammenfassung
The settings for homotopical algebra-categories such as simplicial groups, simplicial rings, A(infinity) spaces, E-infinity 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-infinity algebras, and A(infinity) algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of E-infinity ring spectra to the category of H-infinity 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-infinity(E-+(infinity) Coker J,LK(2)R). (C) 2014 Elsevier Inc. All rights reserved.
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