Higher Tate central extensions via K-theory and infinity-topos theory

TL;DR

This paper introduces a new classification approach for Tate central extensions using K-theory and infinity-topos theory, providing a simpler alternative to classical gerbe-based constructions.

Contribution

It offers a novel classification theorem for torsors over K-theory sheaves via Tate vector bundles, employing infinity-topos language for a more natural framework.

Findings

Classification of torsors over K-theory sheaves

Alternative approach to Tate central extension

Application of infinity-topos theory in K-theory

Abstract

We give a classification theorem of certain geometric objects, called torsors over the sheaf of K-theory spaces, in terms of Tate vector bundles. This allows us to present a very natural and simple, alternative approach to the Tate central extension, which was classically constructed by using the gerbe of determinant theories. We use the language of infinity-topoi as the theoretical framework, since it has well-developed, extended notions of groups, actions, and torsors, which make it possible to regard the sheaf of K-theory spaces as a group object of such kind and to interpret a delooping theorem in K-theory as a classification theorem for torsors over it.