Preorientations of the derived motivic multiplicative group

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.

💡 Research Summary

The paper “Preorientations of the derived motivic multiplicative group” develops a homotopical framework that connects complex topological K‑theory with the derived multiplicative group, first in the classical stable homotopy setting and then in the motivic context. The authors begin by recalling the necessary background on model categories, symmetric spectra, and E∞‑algebras. They adopt the positive stable model structure on symmetric spectra, which ensures that cofibrant‑fibrant objects behave well with respect to smash products and homotopy colimits. Within this setting they reinterpret Lurie’s theorem: the complex K‑theory spectrum KU, equipped with its canonical E∞‑ring structure, represents orientations of the derived multiplicative group Gₘ^der. The key innovation is the introduction of “preorientation”, a weaker notion than a full orientation that only requires a homotopy‑coherent map from the derived group to a given E∞‑algebra. By constructing a natural bijection
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