Synthetic spectra and the cellular motivic category

2018/03/05 by Pstrągowski, Piotr · 4 citations
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.01804

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 ∞-category Syn_E of synthetic spectra based on E is in a precise sense a deformation of the ∞-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 ∞-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.

Cited by

• Real isotropic motivic spectra
• Revisiting the β₁-action on the 3-primary stable homotopy groups of spheres
• Slice spectral sequences through synthetic spectra
• The weight filtration on real singular homology is motivic

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