Stabilizations of $\mathbb{E}_\infty$ Operads and $p$-Adic Stable Homotopy Theory

We study differential graded operads and $p$-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 $\mathbb{E}_\infty$ operads. For example, we construct a stable Barratt-Eccles operad. We develop a homotopy theory of algebras over these stable operads and a theory of (co)homology operations for algebras over these stable operads. We note interesting properties of these operads, such as that, non-equivariantly, in each arity, they have (almost) trivial homology, whereas, equivariantly, these homologies sum to a certain completion of the generalized Steenrod algebra and so are highly non-trivial. We also justify the adjective "stable" by showing that, among other things, the monads associated to these operads are additive in the homotopy coherent, or $\infty$-, sense. We then provide an application of our stable operads to $p$-adic stable homotopy theory. It is well-known that cochains on spaces yield examples of algebras over $\mathbb{E}_\infty$ 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 $\mathbb{E}_\infty$ algebraic structure, cochains on spaces provide algebraic models of $p$-adic homotopy types. We show that, endowed with the algebraic structure encoded by our stable operads, spectral cochains provide algebraic models for $p$-adic stable homotopy types.