We describe a homotopy-theoretic approach to the moduli of @math -algebras of Blanc-Dwyer-Goerss using the @math -category @math of product-preserving presheaves on finite-wedges of positive-dimensional spheres, reproving all of their results in this new setting.
@cite_6: 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).
In his thesis, Mazel-Gee developed the theory of model @math -categories and used it to generalize Goerss-Hopkins theory to the setting of an arbitrary presentable, stable @math -category, see @cite_6 . The current note concerns a simpler problem and is not directly related to Mazel-Gee's work, but the latter's emphasis on homotopy-invariant language and the prominence of product-preserving preshaves provided the author with much of the motivation to work out the case of @math -algebras in detail.