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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 Luries $infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to Rondigs and O stvae rs theorem relating Voevodskys motives with modules over motivic cohomology, and to Garkushas extension of Rondigs and O stvae rs 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 Deglise.
We show that Shipleys detection functor for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological $K$-theory, and semi-topological cob
Recently an algebra of smooth valuations was attached to any smooth manifold. Roughly put, a smooth valuation is finitely additive measure on compact submanifolds with corners which satisfies some extra properties. In this note we initiate a study of
We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding motives (or motivic spectra) of smooth varieties in them; we
We prove a topological invariance statement for the Morel-Voevodsky motivic homotopy category, up to inverting exponential characteristics of residue fields. This implies in particular that SH[1/p] of characteristic p>0 schemes is invariant under pas
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is s