Real isotropic motivic spectra

Abstract

In this paper, we introduce the category of real isotropic motivic spectra, and show that the real realization functor from motivic spectra over $\mathbb{R}$ to classical spectra factors through it. We then describe its cellular subcategory as a one-parameter deformation of the category of spectra, with parameter $\rho$ corresponding to $-1 \in \mathbb{R}^{\times}$, whose special fiber is the derived category of comodules over the dual Steenrod algebra. This leads to an identification of real isotropic cellular spectra with $\mathbb{F}_2$-synthetic spectra, and sheds light on the relation between motivic homotopy theory over $\mathbb{R}$ and $\mathbb{F}_2$-synthetic homotopy theory.