https://hal-univ-artois.archives-ouvertes.fr/hal-02942568Bachmann, TomTomBachmannCalmès, BaptisteBaptisteCalmèsLML - Laboratoire de Mathématiques de Lens - UA - Université d'ArtoisDéglise, FrédéricFrédéricDégliseFasel, JeanJeanFaselØstvær, Paul ArnePaul ArneØstværMilnor-Witt MotivesHAL CCSD2020[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Calmès, Baptiste2020-09-18 09:23:582021-11-03 06:19:352020-09-18 09:23:58enPreprints, Working Papers, ...1We develop the theory of Milnor-Witt motives and motivic cohomology. Compared to Voevodsky's theory of motives and his motivic cohomology, the first difference appears in our definition of Milnor-Witt finite correspondences, where our cycles come equipped with quadratic forms. This yields a weaker notion of transfers and a derived category of motives that is closer to the stable homotopy theory of schemes. We prove a cancellation theorem when tensoring with the Tate object, we compare the diagonal part of our Milnor-Witt motivic cohomology to Minor-Witt K-theory and we provide spectra representing various versions of motivic cohomology in the $\mathbb{A}^1$-derived category or the stable homotopy category of schemes.