Summary: n-Motivic Sheaves
This talk is based on our joint paper  with L. Barbieri-Viale. We fix a ground
field k which we assume, for simplicity, to be of characteristic zero. Also for
simplicity, we will work with rational coefficients. In the sequel, motivic sheaf is a
shorthand for homotopy invariant sheaf with transfers , i.e., a motivic sheaf F
is an additive contravariant functor from the category of smooth correspondences
Cor(k) (see [3, Def. 1.5]) to the category of Q-vector spaces such that:
(a) for every smooth k-scheme X, F(X) F(A1
X) is invertible.
(b) the restriction of F to the category Sm/k of smooth k-schemes is a Nis-
nevich (or equivalently, an ´etale) sheaf with transfers.
If F satisfy (b) but not necessarily (a), we call it a sheaf with transfers. The
category of sheaves with transfers will be denoted by Str(k). We denote HI(k) its
full subcategory of motivic sheaves. The obvious inclusion admits a left adjoint
h0 : Str(k) HI(k). It follows from [3, Th. 22.3] that h0 is the given by the
Nisnevich sheaf of the associated homotopy invariant presheaf with transfers. In
particular, HI(k) is an abelian category and the inclusion HI(k) Str(k) is exact.
In fact, there is a natural t-structure on Voevodsky's category DMeff (k) whose
heart is canonically equivalent to HI(k). This gives a hint why motivic sheaves