Synthetic spectra and the cellular motivic category

DOI10.1007/S00222-022-01173-2arXiv1803.01804MaRDI QIDQ6298622

Piotr Pstrągowski

Publication date: 5 March 2018

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 $Sy{n}_E$ 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.

Mathematics Subject Classification ID

Spectra with additional structure ((E_infty), (A_infty), ring spectra, etc.) (55P43) Adams spectral sequences (55T15) Motivic cohomology; motivic homotopy theory (14F42) ((infty,1))-categories (quasi-categories, Segal spaces, etc.); (infty)-topoi, stable (infty)-categories (18N60)

This page was built for publication: Synthetic spectra and the cellular motivic category

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6298622)