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). We prove a homological mirror symmetry equivalence between an @math -brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a @math -brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the @math -model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. 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).