Integral Models for Spaces via the Higher Frobenius

Abstract

We give a fully faithful integral model for spaces in terms of $\mathbb{E}_{\infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete $\mathbb{E}_{\infty}$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$ is the data of its Spanier-Whitehead dual as an $\mathbb{E}_{\infty}$-ring together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen's $Q$-construction acts on the $\infty$-category of $\mathbb{E}_{\infty}$-rings with "genuine equivariant multiplication," which we call global algebras. The second is a "pre-group-completed" variant of algebraic $K$-theory which we call partial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of $\mathbb{F}_p$ up to $p$-completion.