Reconstructing rational stable motivic homotopy theory

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy-Levine-Panin, a theorem of Cisinski-Deglise and a version of the Roendigs-Ostvaer theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor-Witt correspondences in the sense of Calmes-Fasel.Comment: This is the final version, accepted in Compositio Mathematic

On Modules Over Motivic Ring Spectra

In this note, we provide an axiomatic framework that characterizes the stable $\infty$-categories that are module categories over a motivic spectrum. This is done by invoking Lurie's $\infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to R\"ondigs and \O stv\ae r's theorem relating Voevodsky's motives with modules over motivic cohomology, and to Garkusha's extension of R\"ondigs and \O stv\ae r's result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calm\`es and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and D\'eglise.Comment: 18 pages, v3. Referee report incorporated, Cor 5.8 made conditional on an \infty-categorical construction of E-correspondences, various minor changes. Submitted. Comments welcome

The triangulated categories of framed bispectra and framed motives

An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{nis}^{fr}(k)$ and $SH_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{eff}(k)$ respectively. We also recover $SH(k)$ and $SH^{eff}(k)$ as the triangulated category of framed motivic spectral functors $SH_{S^1}^{fr}[\mathcal Fr_0(k)]$ and the triangulated category of framed motives $\mathcal {SH}^{fr}(k)$ respectively constructed in the paper

Rational enriched motivic spaces

Rational enriched motivic spaces are introduced and studied in this thesis to provide new models for connective and very effective motivic spectra with rational coefficients. We first study homological algebra for Grothendieck categories of functors enriched in Nisnevich sheaves with specific transfers A. Following constructions of Voevodsky for triangulated categories of motives and framed motivic-spaces, we introduce and study motivic structures on unbounded chain complexes of enriched functors yielding two new models of the triangulated category of big motives with A-tranfers DMA. We next dene enriched motivic spaces as certain enriched functors of simplicial A-sheaves. We then use the properties of enriched motivic spaces and the above reconstruction results to recover SH(k)>0,Q and SHveff(k)Q