Lifting homotopy \(T\)-algebra maps to strict maps
type: Item
instance of: scholarly article
title: Lifting homotopy \(T\)-algebra maps to strict maps
zbMATH Open document ID: 1306.55006
author: Niles Johnson
author: Justin Noel
published in: Advances in Mathematics
publication date: 2014-08-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.1511
reviewed by: Martin Frankland
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55S35
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 55P43
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 55T05
Mathematics Subject Classification ID: 55Q50
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 6334313
zbMATH Keywords: monads
zbMATH Keywords: homotopy theory
zbMATH Keywords: spectral sequences
zbMATH Keywords: obstruction theory
zbMATH Keywords: operads
MaRDI profile type: Publication
OpenAlex ID: W2169445645
arXiv ID: 1301.1511
cites work: Q5200272
cites work: Q3164468
cites work: Isogenies of formal group laws and power operations in the cohomology theories \(E_ n\)
cites work: Elliptic spectra, the Witten genus and the theorem of the cube
cites work: The sigma orientation is an H [infinity] map
cites work: Topological Hochschild homology and cohomology of \(A_{\infty }\) ring spectra
cites work: Axiomatic homotopy theory for operads
cites work: The Boardman-Vogt resolution of operads in monoidal model categories
cites work: Reedy categories and the \(\varTheta\)-construction
cites work: Generalized André-Quillen cohomology
cites work: Homotopical resolutions associated to deformable adjunctions
cites work: Q4304740
cites work: Homotopy spectral sequences and obstructions
cites work: C ∞ -Schemes
cites work: Q4716283
cites work: Q4664146
cites work: Simplicial homotopy theory
cites work: Q4790109
cites work: The K-theory of some more well-known spaces
cites work: Q4220599
cites work: Symmetric spectra
cites work: For complex orientations preserving power operations, \(p\)-typicality is atypical
cites work: Q4215656
cites work: Q3934579
cites work: Suspension Spectra and Homology Equivalences
cites work: Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces
cites work: FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES
cites work: Higher Topos Theory (AM-170)
cites work: Equivariant orthogonal spectra and 𝑆-modules
cites work: The geometry of iterated loop spaces
cites work: Q5687557
cites work: Q3997818
cites work: Homotopical algebra
cites work: Q5642708
cites work: The Morava K-Theories of Eilenberg-MacLane Spaces and the Conner-Floyd Conjecture
cites work: Q4218781
cites work: Galois extensions of structured ring spectra. Stably dualizable groups
cites work: Stable homotopical algebra and Γ-spaces
cites work: Stable homotopy of algebraic theories
cites work: Morava \(E\)-theory of symmetric groups
DOI: 10.1016/J.AIM.2014.07.002
review text: 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…
description: scientific article; zbMATH DE number 6334313