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Lifting homotopy T-algebra maps to strict maps
The settings for homotopical algebra---categories such as simplicial groups,
simplicial rings, spaces, ring spectra, etc.---are often
equivalent to categories of algebras over some monad or triple . In such
cases, is acting on a nice simplicial model category in such a way that
descends to a monad on the homotopy category and defines a category of homotopy
-algebras. In this setting there is a forgetful functor from the homotopy
category of -algebras to the category of homotopy -algebras.
Under suitable hypotheses we provide an obstruction theory, in the form of a
Bousfield-Kan spectral sequence, for lifting a homotopy -algebra map to a
strict map of -algebras. Once we have a map of -algebras to serve as a
basepoint, the spectral sequence computes the homotopy groups of the space of
-algebra maps and the edge homomorphism on 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 -term to be calculable
in terms of Quillen cohomology groups.
We provide worked examples in -spaces, -spectra, rational
algebras, and algebras. Explicit calculations, connected to rational
unstable homotopy theory, show that the forgetful functor from the homotopy
category of ring spectra to the category of 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
.Comment: 45 pages. Substantial revision. To appear in Advances in Mathematic
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