Lifting homotopy \(T\)-algebra maps to strict maps
In homotopy theory, one often considers homotopical information that comes in more or less structured versions. For example, there is a forgetful functor from \(E_{\infty}\) ring spectra to \(H_{\infty}\) ring spectra. Another example is the forgetful functor from (strict) \(G\)-spaces to \(G\)-objects in the homotopy category of spaces, for a group \(G\). The paper under review provides general machinery for studying such forgetful functors and the associated problem of lifting a less structured map to a more structured one.NEWLINENEWLINELet \(C\) be a nice simplicial model category and \(T\) a nice simplicial monad on \(C\), and let \(C_T\) denote the category of \(T\)-algebras, with forgetful functor \(U : C_T \to C\). The monad \(T\) induces a monad \(hT\) on the homotopy category \(ho C\). For example, if \(T\) is the monad on spectra whose algebras are \(E_{\infty}\) ring spectra, then \(hT\)-algebras are \(H_{\infty}\) ring spectra.NEWLINENEWLINEThe first main result (Theorem A) is a spectral sequence, called the \(T\)-algebra spectral sequence, exhibiting the forgetful maps \(ho C_T(X,Y) \to (ho C)_{hT}(UX,UY) \to ho C(UX,UY)\) as edge morphisms. The differential \(d_1\) provides an obstruction to lifting a homotopy class \([f] : UX \to UY\) to a map of \(hT\)-algebras. Given a map of \(T\)-algebras \(X \to Y\) as basepoint, the spectral sequence converges to the homotopy groups of the (derived) mapping space of \(T\)-algebra maps \(X \to Y\). In that case, the differentials \(d_r\) for \(r \geq 2\) provide obstructions to lifting a map of \(hT\)-algebras to a map of \(T\)-algebras. The authors study various properties of the spectral sequence and provide contexts in which the technical assumptions are satisfied, so that the spectral sequence can be used in practice.NEWLINENEWLINEThe second main result (Theorem B) is an identification, under suitable hypotheses, of the \(E_2\) term of the spectral sequence with Quillen cohomology groups. This is a key result for computations.NEWLINENEWLINESection 5 contains several applications:NEWLINENEWLINE(5.1) \(G\)-spaces and \(G\)-spectra.NEWLINENEWLINE(5.2.1) \(A_{\infty}\) algebras in \(Hk\)-modules for \(k\) a field.NEWLINENEWLINE(5.2.2) \(E_{\infty}\) algebras in \(Hk\)-modules for \(k\) a field of characteristic \(0\). Notably, the authors show that the forgetful functor from the homotopy category of \(E_{\infty}\) ring spectra to \(H_{\infty}\) ring spectra is neither full nor faithful (Examples 5.12 and 5.9).NEWLINENEWLINE(5.3) \(E_{\infty}\) ring spectra in the \(K(2)\)-local category.NEWLINENEWLINEThe paper is thorough and well written. The machinery developed by the authors lends itself to further applications.
scientific article; zbMATH DE number 6334313
scientific article