We present an abstract version of Goerss-Hopkins theory in the setting of a prestable @math -category equipped with a suitable periodicity operator. In the case of the @math -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.
To give a different example, Goerss-Hopkins towers were also used by the first author to prove that at large primes, there is an equivalence @math of homotopy categories between @math -local spectra and differential @math -comodules @cite_0 . The proof of that result proceeds by comparing topological and algebraic towers and so crucially depends on the additional generality provided in this note.
@cite_0
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We show that if @math is a @math -local Landweber exact homology theory of height @math and @math , then there exists an equivalence @math between homotopy categories of @math -local spectra and differential @math -comodules, generalizing Bousfield's and Franke's results to heights @math . Goerss--Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model @math -categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable @math -category (such as that of equivariant spectra or of motivic spectra). Author(s): Mazel-Gee, Aaron | Advisor(s): Teichner, Peter | Abstract: We develop a theory of model ∞-categories -- that is, of model structures on ∞-categories -- which provides a robust theory of resolutions entirely native to the ∞-categorical context. Using model ∞-categories, we generalize Goerss--Hopkins obstruction theory from spectra to an arbitrary (presentably symmetric monoidal stable) ∞-category. We give a sample application of this generalized obstruction theory in the setting of motivic homotopy theory, where we construct E_∞ structures on the motivic Morava E-theories and compute their automorphism spaces (as E_∞ algebras). Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The @math -model structure on @math Equivariant @math -type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho @math and @math Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and @math -categories The category of orthogonal @math -spectra over @math Model structures for parametrized @math -spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation.