To any Adams-type homology theory we associate a notion of a synthetic spectrum, this is a spherical sheaf on the site of finite spectra with projective E-homology. We show that the ∞-category Syn E of synthetic spectra based on E is symmetric monoidal, stable, and that it is in a precise sense a deformation of Hovey's stable homotopy theory of E * E-comodules whose generic fibre is the ∞-category of spectra. It follows that the Adams spectral sequence in Syn E interpolates between the topological and algebraic Adams spectral sequences.We describe a symmetric monoidal functor Θ * : Sp C → Syn ev M U from the ∞-category of cellular motivic spectra over Spec(C) into an even variant of synthetic spectra based on M U and show that Θ induces an equivalence between the ∞-categories of p-complete objects for all primes p. This establishes a purely topological model for the p-complete cellular motivic category and gives an intuitive explanation of the "Cτ -philosophy" of Gheorghe, Isaksen, Wang and Xu. 6.1. Cellularity 59 6.2. The synthetic dual Steenrod algebra 60 7. Comparison with the cellular motivic category 64 7.1. Cellular motivic category 64 7.2. Finite M GL-projective motivic spectra 66 7.3. Cellular motivic category as spherical sheaves 69 7.4. Homotopy of p-complete finite motivic spectra 72 7.5. A topological model for the p-complete cellular motivic category 74 Appendix A. Sheaves 76 A.1. Grothendieck pretopologies 76 A.2. Hypercompleteness and hypercovers 79 References 82

To any well-behaved homology theory we associate a derived 8-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration. 7.1. The algebraic model 87 7.2. Splittings of abelian categories and the Bousfield functor 89 7.3. Bousfield adjunction 91 7.4. The truncated thread monad 94 7.5. Proof of Franke's conjecture 99 8. Applications of the conjecture 101 8.1. Modules over ring spectra 102 8.2. Chromatic algebraicity 105 8.3. Diagram 8-categories 105 Appendix A. Comodules and quotients of abelian categories 110 References 112

We present an abstract version of Goerss-Hopkins theory in the setting of a prestable ∞-category equipped with a suitable periodicity operator. In the case of the ∞category of synthetic spectra, this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.

No abstract

As a consequence of the algebraicity of chromatic homotopy at large primes, we show that the Hopkins' Picard group of the K(n)-local category coincides with the algebraic one when 2p .22]. The latter Hopf algebroid can be described explicitly as E ∨ * E ≃ map c (G n , E * ), the space of continuous functions on the Morava stabilizer group, with structure maps induced from the action of) into the algebraic Picard group, given by isomorphisms classes of invertible comodules.The algebraic Picard group can be expressed in terms of cohomology of the Morava stabilizer group; to do so, one observes that an invertible E ∨ * E-comodule is the same as an invertible E * -module equipped with a compatible continuous action of G n . Since E 0 is a regular local ring, any such module is free of rank one, and so we have a short exact sequence) is the subgroup of those invertible modules which are concentrated in even degrees. Since E * is 2-periodic, any such module is determined by its degree zero part, which yields an isomorphism Pic 0 (E ∨ * E) ≃ H 1 c (G n , E × 0 ) by standard considerations [GHMR14].