Synthetic spectra are (usually) cellular

TL;DR

This paper explores the structure of synthetic spectra generated by connective ring spectra, showing their equivalence to modules over filtered ring spectra, which advances understanding of their algebraic and categorical properties.

Contribution

It establishes that the category of $E$-synthetic spectra is generated by bigraded spheres and is equivalent to modules over a filtered ring spectrum, revealing new structural insights.

Findings

Synthetic spectra are generated by bigraded spheres.

The category of synthetic spectra is equivalent to modules over a filtered ring spectrum.

Provides a new perspective on the algebraic structure of synthetic spectra.

Abstract

If $E$ is a connective ring spectrum, then Pstragowski's category $Syn_E$ of $E$-synthetic spectra is generated by the bigraded spheres $S^{i,j}$. In particular, it is equivalent to the category of modules over a filtered ring spectrum.