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 space \(X\) 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.