We compare two approaches to the homotopy theory of ∞-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and ∞-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Γ. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.

We compare two models for ∞-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for ∞-operads are equivalent -for instance, it follows that the homotopy theory of Lurie's ∞-operads is equivalent to that of dendroidal sets and that of simplicial operads.

Introduction vii Chapter 1. Main results Chapter 2. Constructing n-excisive approximations Chapter 3. Another construction of polynomial approximations Chapter 4. Coalgebras in stable ∞-operads 4.1. Truncations of ∞-operads 4.2. Coalgebras in a corepresentable ∞-operad 4.3. Coalgebras in an n-truncated stable ∞-operad Chapter 5. The space of Goodwillie towers 5.1. The Tate diagonal 5.2. Constructing n-stages 5.3. A classification of n-stages 5.4. The case of vanishing Tate constructions Chapter 6. Examples 6.1. Divided power coalgebras and Koszul duality 6.2. Rational homotopy theory 6.3. Spaces with homotopy groups in a finite range 6.4. Spaces and Tate coalgebras 6.5. Further remarks on the Goodwillie tower of the homotopy theory of spaces Appendix A. Compactly generated ∞-categories Appendix B. Some facts from Goodwillie calculus Appendix C. Truncations C.1. The homotopy theory of truncated ∞-operads C.2. Truncations of stable ∞-operads C.3. A cobar construction for stable ∞-operads C.4. Coalgebras Bibliography iii

Abstract. For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with the covariant model structures. Also, we show that versions of Quillen's Theorems A and B for ∞-categories easily follow.

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in vn-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this vn-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T (n)-local spectra. We also compare it to the homotopy theory of commutative coalgebras in T (n)-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity. Contents 1. Introduction Acknowledgments 2. Main results 3. The v n -periodic homotopy theory of spaces 3.1. The ∞-category L f n 3.2. The ∞-category M f n 3.3. The stabilization of M f n 3.4. The Bousfield-Kuhn functor 4. Lie algebras in T (n)-local spectra 4.1. Operads and cooperads of T (n)-local spectra 4.2. The functor ΦΘ 4.3. The monad structure of ΦΘ 4.4. Some applications 4.5. A variation for K(n)-local homotopy theory 5. The Goodwillie tower of M f n 5.1. Stabilizations and Goodwillie towers 5.2. The Goodwillie tower of the Bousfield-Kuhn functor Appendix A. Dual Goodwillie calculus Appendix B. A nilpotence lemma for differentiation B.1. Statement of the lemma B.2. Some preliminaries on nilpotence B.3. Proof Appendix C. CoalgebrasReferences Bousfield-Kuhn functor and topological André-Quillen homology. I have benefitted much from reading their work, as well as from several stimulating talks by and conversations with Mark Behrens. Theorem 2.6 (the comparison with Lie algebras) offers a different (and sharper) perspective on v n -periodic homotopy theory, which builds on joint work with Rosona Eldred, Akhil Mathew, and Lennart Meier [19] carried out at the Hausdorff Research Institute for Mathematics. I wish to thank my collaborators for an inspiring semester and the Institute for its hospitality and excellent working conditions. Moreover, I thank Greg Arone and Lukas Brantner for useful conversations relating to this paper. A large intellectual debt is owed to