Milnor-Witt homotopy sheaves and Morel generalized transfers

Abstract

We explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. Moreover, we study the relations between (contracted) homotopy sheaves, sheaves with Morel generalized transfers and Milnor-Witt homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets Milnor-Witt transfers and use these results to discuss the conservativity conjecture in motivic homotopy theory due to Bachmann and Yakerson.