Synthetic spectra and the cellular motivic category

Abstract

To an Adams-type homology theory we associate the notion of a synthetic spectrum; this is a product-preserving sheaf on the site of finite spectra with projective E-homology. We show that the $$\infty $$ -category 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 the $$\infty $$ -category of cellular motivic spectra over $$\text {Spec}({\mathbb {C}})$$ into an even variant of synthetic spectra based on $$\textrm{MU}$$ 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.