Topological rigidification of schemes

TL;DR

The paper introduces a universal construction for schemes that simplifies their structure by removing nontrivial universal homeomorphisms, enabling a categorical framework aligned with Voevodsky's topos.

Contribution

It provides a new categorical method to invert universal homeomorphisms for schemes, connecting algebraic geometry with higher topos theory.

Findings

Constructs a universal reduced scheme from any scheme via a universal homeomorphism.

Embeds the resulting category into Voevodsky's h infinity-topos.

Establishes a foundation for formal inversion of universal homeomorphisms in algebraic geometry.

Abstract

We show that any of a large class of schemes receives a universal homeomorphism from a reduced scheme that in turn receives no nontrivial universal homeomorphism from any other reduced scheme. This construction serves as a categorical input for the formal inversion of universal homeomorphisms; the result is an infinity-category that embeds as a full subcategory of the h infinity-topos of Voevodsky.