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.
A different approach to Goerss-Hopkins obstruction theory is present in the work of Mazel-Gee @cite_27 . Compared to ours, the approach of Mazel-Gee is closer in spirit to the original works of Goerss and Hopkins, generalizing the theory to a setting of an arbitrary presentable @math -category through the use of model @math -categories.
@cite_27
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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).