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
2904500875
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).