Spherical Witt vectors and integral models for spaces

TL;DR

This paper introduces a new construction of spherical Witt vectors, extends them to nonconnective objects, and explores their applications in integral models for spaces and schematization, connecting to recent advances in algebraic topology.

Contribution

It provides a novel construction of spherical Witt vectors using synthetic spectra, extending their applicability and linking them to integral cochains and derived $mbda$-rings.

Findings

Construction of spherical Witt vectors via synthetic spectra.

Fully faithful functor from spaces to derived $mbda$-rings.

Connections to schematization and recent work of Horel and Kubrak-Shuklin-Zakharov.

Abstract

We give a new construction of the spherical Witt vector functor of Lurie and Burklund-Schlank-Yuan and extend it to nonconnective objects using synthetic spectra and recent work of Holeman. The spherical Witt vectors are used to build spherical versions of perfect $\lambda$-rings and to motivate new results in Grothendieck's schematization program, building on work of Ekedahl, Kriz, Mandell, Lurie, Quillen, Sullivan, To\"en, and Yuan. In particular, there is an $\infty$-category of perfect derived $\lambda$-rings with trivializations of the Adams operations $\psi^p$ for all $p$ such that the functor sending a space $X$ to its integral cochains on $X$, viewed as such a derived $\lambda$-ring, is fully faithful on a large class of nilpotent spaces. Our theorem is closely related to recent work of Horel and Kubrak-Shuklin-Zakharov. Finally, we answer two questions of Yuan on spherical cochains.