The aim of this paper is to study convergence of Bousfield-Kan completions with respect to the 1-excisive approximation of the identity functor and exotic convergence of the Taylor tower of the identity functor, for algebras over operads in spectra centered away from the null object. In Goodwillie's homotopy functor calculus, being centered away from the null object amounts to doing homotopy theory and functor calculus in the retractive setting.

We prove that the stabilization (resp. iterated suspension) functor participates in a derived adjunction comparing pointed spaces with certain (highly homotopy coherent) homotopy coalgebras, in the sense of Blumberg-Riehl, that is a Dwyer-Kan equivalence after restriction to 1-connected spaces, with respect to the associated enrichments. A key ingredient of our proof, of independent interest, is a higher stabilization theorem (resp. higher Freudenthal suspension theorem) for pointed spaces that provides strong estimates for the uniform cartesian-ness of certain cubical diagrams associated to iterating the space level stabilization map (resp. Freudenthal suspension map)—these technical results provide, in particular, new proofs (with strong estimates) of the stabilization and iterated loop-suspension completion results of Carlsson and the subsequent work of Arone-Kankaanrinta, and Bousfield and Hopkins, respectively, for 1-connected spaces; this is the stabilization (resp. Freudenthal suspension) analog of Dundas’ higher Hurewicz theorem.

Working in the Arone-Ching framework for homotopical descent, it follows that the Bousfield-Kan completion map with respect to integral homology is the unit of a derived adjunction. We prove that this derived adjunction, comparing spaces with coalgebra complexes over the associated integral homology comonad, via integral chains, can be turned into a derived equivalence by replacing spaces with the full subcategory of simply connected spaces. In particular, this provides an integral chains characterization of the homotopy type of simply connected spaces.

The aim of this short paper is two-fold: (i) to construct a TQ-localization functor on algebras over a spectral operad O, in the case where no connectivity assumptions are made on the O-algebras, and (ii) more generally, to establish the associated TQ-local homotopy theory as a left Bousfield localization of the usual model structure on O-algebras, which itself is almost never left proper, in general. In the resulting TQ-local homotopy theory, the "weak equivalences" are the TQ-homology equivalences, where "TQ-homology" is short for topological Quillen homology, which is also weakly equivalent to stabilization of O-algebras. More generally, we establish these results for TQ-homology with coefficients in a spectral algebra A. A key observation, that goes back to the work of Goerss-Hopkins on moduli problems, is that the usual left properness assumption may be replaced with a strong cofibration condition in the desired subcell lifting arguments: Our main result is that the TQ-local homotopy theory can be established (e.g., a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are TQ-equivalences.

Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected O-algebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory. This derived Koszul duality result can be thought of as the spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, and the subsequent p-adic and integral work of Goerss and Mandell on cochains and homotopy type---the following are corollaries of our main result: (i) 0-connected O-algebra spectra are weakly equivalent if and only if their TQ-homology spectra are weakly equivalent as derived K-coalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant, then it comes from the TQ-homology spectrum of an O-algebra. We construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups TQ_*(X) of an O-algebra X, and prove that it converges strongly to pi_*(X) when X is 0-connected.

The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a TQ-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here TQ-homology is short for topological Quillen homology. We also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.

We prove that the fundamental adjunction comparing spaces with coalgebra complexes over the associated integral homology comonad, via integral chains, can be turned into an equivalence of homotopy theories by replacing spaces with the full subcategory of 1-connected spaces. This resolves in the affirmative the integral chains problem---to establish an integral chains analog of Quillenu0027s rational chains equivalence in rational homotopy theory; it can be thought of as an integral (dual) analog of Sullivanu0027s rational cochains equivalence in rational homotopy theory.

We prove that the stabilization of spaces functor---the classical construction of associating a spectrum to a pointed space by tensoring with the sphere spectrum---satisfies homotopical descent on objects and morphisms. This is the stabilization analog of the Quillen-Sullivan theory main result that the rational chains (resp. cochains) functor participates in a derived equivalence with certain coalgebra (resp. algebra) complexes, after restriction to 1-connected spaces up to rational equivalence. In more detail, we prove that the stabilization of spaces functor participates in a derived equivalence with certain coalgebra spectra (where the stabilization construction naturally lands), after restriction to 1-connected spaces up to weak equivalence. This resolves in the affirmative the infinite case, involving stabilization and suspension spectra, of a question/conjecture posed by Tyler Lawson on (iterated) suspension spaces almost ten years ago. A key ingredient of our proof, of independent interest, is a higher stabilization theorem for spaces that provides strong estimates for the uniform cartesian-ness of certain cubical diagrams associated to n-fold iterations of the spaces-level stabilization map; this is the stabilization analog of Dundas' higher Hurewicz theorem.

We establish a higher Freudenthal suspension theorem and prove that the derived fundamental adjunction comparing spaces with coalgebra spaces over the homotopical iterated suspension-loop comonad, via iterated suspension, can be turned into an equivalence of homotopy theories by replacing spaces with the full subcategory of 1-connected spaces. This resolves in the affirmative a conjecture of Lawson on iterated suspension spaces; that homotopical descent for iterated suspension is satisfied on objects and morphisms---the corresponding iterated desuspension space can be built as the homotopy limit of a cosimplicial cobar construction encoding the homotopical coalgebraic structure. It also provides a homotopical recognition principle for iterated suspension spaces. In a nutshell, we show that the iterated loop-suspension completion map studied by Bousfield participates in a derived equivalence between spaces and coalgebra spaces over the associated homotopical comonad, after restricting to 1-connected spaces.

Working in the context of symmetric spectra, we prove higher homotopy excision and higher Blakers-Massey theorems, and their duals, for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra).

We establish model category structures on algebras and modules over operads in symmetric spectra and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.

Dmitri Pavlov and Jakob Scholbach have pointed out that part of Proposition 6.3, and hence Proposition 4.28(a), of (1) are incorrect as stated (see (2) for details). In particular, the corresponding Proposition 7.17(a) of (3) is incorrect as stated. While all of the main results of that paper remain unchanged, this necessitates modications to the statements and proofs of a few technical propositions.

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ-completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan's completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes.

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ-completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan's completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes.

We describe and study a (homotopy) completion tower for algebras and left modules over operads in symmetric spectra. We prove that a weak equivalence on topological Quillen homology induces a weak equivalence on homotopy completion, and that for $0$-connected algebras and modules over a $-1$-connected operad, the homotopy completion tower interpolates between topological Quillen homology and the identity functor. By an explicit calculation of its layers, we show that the homotopy completion tower is the precise analog---in the context of algebras and modules over operads---of the Goodwillie tower of the identity functor. As easy consequences of the strong convergence properties of the homotopy completion tower, we prove a Whitehead theorem and a Hurewicz theorem for topological Quillen homology. We also prove a relative Hurewicz theorem that provides conditions under which topological Quillen homology detects $n$-connected maps. We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups; this result should be thought of as an algebras over operads in spectra analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology, and use this to define Quillen homology completion---in the sense of Bousfield-Kan---for algebras and modules over operads. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.

We establish model category structures on algebras and modules over operads and non-Σ operads in monoidal model categories. The results have applications in algebraic topology, stable homotopy theory, and homological algebra.

We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.