A presheaf is -local if the natural map is an equivalence for all . We write for the full subcategory of -local presheaves, and for the presheaves that are both Nisnevich local and -local.
Therefore we get a square of accessible localizations
We also write the composite as .
is not a sheafification functor. Thus, a priori
, the only nice property of it we know is that it is a left adjoint. To remedy for this, we describe an explicit construction of
, and then observe that
The functor is better known as .
Define the “affine -simplex” by
This forms a cosimplicial scheme in the usual way.
For , we define by
It is then straightforward to check that
preserves finite products. Hence so does .
Let be a (Nisnevich-)locally trivial -bundle. Then is an -equivalence in .
We claim that . Here we are working with pointed objects so that suspensions make sense.
Indeed, we have a Zariski open cover of given by
Since this is in particular an elementary distinguished square, it is also a pushout in . Now apply to this diagram, which preserves pushouts since it is a left adjoint. Since the claim follows.
If we “replace” only one of the 's with , we see that this also says
In general, we have
It is convenient to introduce the following notation:
We define whenever it makes sense.