  Strongly (respectively strictly) A1-invariant sheaves are foundational for
motivic homotopy theory over fields. They are sheaves of (abelian) groups on
the Nisnevich site of smooth varieties over a field k, with the property that
their zeroth and first Nisnevich cohomology sets (respectively all Nisnevich
cohomology groups) are invariant under replacing a variety X by the affine line
over X. A celebrated theorem of Fabien Morel states that if the base field k is
perfect, then any strongly A1-invariant sheaf of abelian groups is
automatically strictly A1-invariant.
  The aim of these lecture notes is twofold: (1) provide a complete proof if
this result, and (2) outline some of its applications.
