Real isotropic motivic spectra

TL;DR

This paper introduces the category of real isotropic motivic spectra, showing its relation to classical spectra and connecting motivic homotopy theory over  with -synthetic homotopy theory, revealing new structural insights.

Contribution

It defines the real isotropic motivic spectra category and describes its cellular subcategory as a deformation of spectra, linking motivic and synthetic homotopy theories.

Findings

Real isotropic motivic spectra factor the real realization functor.

Cellular subcategory is a deformation of the spectra category.

Identifies real isotropic cellular spectra with -synthetic 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.