Lifting homotopy T-algebra maps to strict maps

摘要

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.