Stabilizations of 𝔼∞ Operads and 𝕡-Adic Stable Homotopy Theory

We study differential graded operads and p p -adic stable homotopy theory. We first construct a new class of differential graded operads, which we call the stable operads. These operads are, in a particular sense, stabilizations of E ∞ E_ operads. We provide an application of these stable operads to p p -adic stable homotopy theory. It is well-known that cochains on spaces yield examples of algebras over E ∞ E_ operads. We show that, in the stable case, cochains on spectra yield examples of algebras over our stable operads. Moreover, a result of Mandell says that, endowed with the E ∞ E_ algebraic structure, cochains on spaces provide algebraic models of p p -adic homotopy types. We show that, endowed with the algebraic structure encoded by our stable operads, spectral cochains provide algebraic models for p p -adic stable homotopy types.