Descent, Affine Representability and Classifying Space in A1-Homotopy Theory

This is my master thesis in Warwick, supervised by Marco Schlichting.

Abstract
We relate hyperdescent condition for simplicial presheaves, with hyperdescent in $$\infty$$-topos and descent for stacks. One consequence is that the classifying space of a stack satises hyperdescent which simplies existing proofs in some cases. We show an algebraic topological approach to some properties of singular constructions for sites with interval. Using results of affine representability and A1-algebraic topology over a field, we show that the A1-homotopy theory is not a model topos. As an extension of this result, we prove that an A1-local monoid is strongly A1-invariant if and only if its 0-th Nisnevich homotopy sheaf is strongly A1-invariant. This can be used for calculation of A1-loop space and we apply it to BBGm. Finally we present some calculations of A1-homotopy sheaves of BGLn and BSLn and in particular
the first a few A1-homotopy sheaves of BGL2.