Norms and Transfers in Motivic Homotopy Theory

Abstract

In this article, we establish the compatibility between norms and transfers in motivic homotopy theory. More precisely, we construct norm functors for motivic spaces equipped with various flavours of transfer. This yields a norm monoidal refinement of the infinite ${\mathbb{P}}^1$-delooping machine of Elmanto-Hoyois-Khan-Sosnilo-Yakerson. We apply this refinement to construct a normed algebra structure on the very effective Hermitian $\mathrm{K}$-theory spectrum.

53 pages, comments welcome! Updates in Version 2: reworked exposition and organization; added further discussion of norm monoidal categories