Notes and Remarks on Unstable Motivic Homotopy Theory

I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.

Affine representability results: 

For remarks on section 2.3 and Lemma 2.3.2 of [2] (also quoted as Proposition 5.5 and Proposition 5.6 in [1]), we take the approach of stack and descent.  By relating hyperdescent condition for simplicial presheaves with descent for stacks [3], we show that how this implies classifying space of a stack satisfies hyperdescent.

After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.

Exercises on the construction of motivic homotopy theory: 

I also have some informal notes about the exercises in [1]. The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.



[1] Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry95, 305-370.

[2] Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology22(2), 1181-1225.

[3] Jardine, J. F. (2015). Local homotopy theory. Springer.

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

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

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.