Synthetic spectra and the cellular motivic category

TL;DR

This paper introduces synthetic spectra associated with Adams-type homology theories, showing they form a deformation of spectra into sheaves over an algebraic stack, and connects cellular motivic spectra over complex numbers to synthetic spectra, revealing chromatic homotopy theory's role.

Contribution

It constructs synthetic spectra as a deformation of spectra into sheaves over an algebraic stack and relates cellular motivic spectra to synthetic spectra via a symmetric monoidal functor.

Findings

Synthetic spectra encode the $E$-based Adams spectral sequence.

The $ategory$ of synthetic spectra is a deformation of spectra into sheaves.

Cellular motivic spectra over complex numbers are equivalent to synthetic spectra in a $p$-complete sense.

Abstract

To an Adams-type homology theory we associate a notion of a synthetic spectrum, this is a product-preserving sheaf on the site of finite spectra with projective $E$-homology. We prove that the $\infty$-category $Syn_{E}$ of synthetic spectra based on $E$ is in a precise sense a deformation of the $\infty$-category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $E$-based Adams spectral sequence. We describe a symmetric monoidal functor from cellular motivic spectra over the complex numbers into an even variant of synthetic spectra based on $MU$ and show that it induces an equivalence between the $\infty$-categories of $p$-complete objects for all primes $p$. In particular, it follows that the $p$-complete cellular motivic category can be described purely in terms of chromatic homotopy theory.