We provide an axiomatic framework that characterizes the stable
-categories
that are module categories over a motivic spectrum. This is done by invoking Lurie’s
-categorical
version of the Barr–Beck theorem. As an application, this gives an alternative
approach to Röndigs and Østvær’s theorem relating Voevodsky’s motives with
modules over motivic cohomology and to Garkusha’s extension of Röndigs and
Østvær’s result to general correspondence categories, including the category of
Milnor–Witt correspondences in the sense of Calmès 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
Déglise.
