Preorientations of the derived motivic multiplicative group

TL;DR

This paper proves that topological complex K-theory represents orientations of the derived multiplicative group and extends this result to the motivic setting, establishing key model structures and adjunctions.

Contribution

It provides a model category proof of Lurie's theorem and generalizes it to the motivic context, introducing new model structures and adjunctions.

Findings

Topological K-theory represents orientations of the derived multiplicative group.

The result is extended to the motivic setting.

New model structures and Quillen adjunctions are established.

Abstract

We provide a proof in the language of model categories and symmetric spectra of Lurie's theorem that topological complex $K$-theory represents orientations of the derived multiplicative group. Then we generalize this result to the motivic situation. Along the way, a number of useful model structures and Quillen adjunctions both in the classical and in the motivic case are established.