3 Homotopy sheaves
Naively, one would like to define the motivic homotopy groups of as . This is an abelian group. We can do better than that, and produce as a sheaf.
Let . We define to be the Nisnevich sheafification of the presheaf of sets
If and , define to be the Nisnevich sheafification of the presheaf of groups
In general, if , we define . Finally, we define
If is a fiber sequence in , then there is a long exact sequence
of Nisnevich sheaves.
The forgetful functor
is a right adjoint, hence preserves fiber sequences. So fiber sequences in
are computed objectwise. Then note that sheafification is exact.
Here it is essential that we did not include in the definition of , since is not exact.
If , we say is -connected if the canonical map induces an isomorphism of sheaves .
Given , to check if is -connected, it turns out it suffices to check that is trivial.
Let . Then the canonical map
induces an epimorphism
is an isomorphism, it suffices to show that
is always an epimorphism. This follows by inspection.
The final property of we note is that over a perfect field, is unramified. Roughly speaking, it says
is injective for any . We cannot exactly say this because is in general not smooth over .