UC Berkeley

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).